Mathematics 10 — Exercise Review · Q2 (ii)

Full question

Solve by factorization, completing the square AND the quadratic formula.

Step-by-step solution

1. Method: factor.

2. Substitute the function expressions, keeping the brackets.

   $$-(-2\times x^2+x+10)-0$$

3. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$2\times x^2-x-10$$

4. Write EVERY distributed product before collecting terms.

   $$-10-x+2\times x^2-0$$

5. Calculate this individual product, including its sign and powers.

   $$-10=-10$$

6. Calculate this individual product, including its sign and powers.

   $$-x=-x$$

7. Calculate this individual product, including its sign and powers.

   $$2\times x^2=2\times x^2$$

8. Calculate this individual product, including its sign and powers.

   $$\left(-1\right)\times 0=0$$

9. These terms have the same variable part. Add ONLY their coefficients.

   $$-10+0=-10$$

10. These terms have the same variable part. Add ONLY their coefficients.

    $$-1=-1$$

11. These terms have the same variable part. Add ONLY their coefficients.

    $$2=2$$

12. Put the simplified groups together.

    $$2\times x^2-x-10$$

13. Collect terms with the same power and simplify their coefficients.

    $$2\times x^2-x-10$$

14. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$2\times x^2-x-10=0$$

15. Split the middle coefficient into two numbers whose sum is b and whose product is a·c.

    $$(-5+4=-1,\left(-5\right)\times 4=-20)$$

16. Rewrite the middle term using those two parts.

    $$2\times x^2-5\times x+4\times x-10=0$$

17. Group the first two terms and the last two. Take out the common factor from each group.

    $$x\times \left(2\times x-5\right)+2\times \left(2\times x-5\right)=0$$

18. The common bracket can now be factored out.

    $$\left(x+2\right)\times \left(2\times x-5\right)=0$$

19. A product is zero only if at least one factor is zero. Set each linear factor equal to zero.

    $$(x+2=0,2\times x-5=0)$$

20. Take this factor equal to zero.

    $$x+2=0$$

21. Subtract the constant from both sides.

    $$x+2-2=0-2$$

22. Simplify both sides.

    $$x=-2$$

23. Divide both sides by the coefficient of the variable.

    $$x=-2\times \frac{1}{1}$$

24. Reduce the fraction to obtain this root.

    $$x=-2$$

25. Take this factor equal to zero.

    $$2\times x-5=0$$

26. Subtract the constant from both sides.

    $$2\times x-5+5=0+5$$

27. Simplify both sides.

    $$2\times x=5$$

28. Divide both sides by the coefficient of the variable.

    $$x=\frac{5}{2}$$

29. Reduce the fraction to obtain this root.

    $$x=\frac{5}{2}$$

30. Substitute this root into the original equation before simplifying.

    $$0=0$$

31. Calculate both sides; their equal values confirm the root.

    $$0=0$$

32. Substitute this root into the original equation before simplifying.

    $$0=0$$

33. Calculate both sides; their equal values confirm the root.

    $$0=0$$

34. Method: square.

35. Substitute the function expressions, keeping the brackets.

    $$-(-2\times x^2+x+10)-0$$

36. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

    $$2\times x^2-x-10$$

37. Write EVERY distributed product before collecting terms.

    $$-10-x+2\times x^2-0$$

38. Calculate this individual product, including its sign and powers.

    $$-10=-10$$

39. Calculate this individual product, including its sign and powers.

    $$-x=-x$$

40. Calculate this individual product, including its sign and powers.

    $$2\times x^2=2\times x^2$$

41. Calculate this individual product, including its sign and powers.

    $$\left(-1\right)\times 0=0$$

42. These terms have the same variable part. Add ONLY their coefficients.

    $$-10+0=-10$$

43. These terms have the same variable part. Add ONLY their coefficients.

    $$-1=-1$$

44. These terms have the same variable part. Add ONLY their coefficients.

    $$2=2$$

45. Put the simplified groups together.

    $$2\times x^2-x-10$$

46. Collect terms with the same power and simplify their coefficients.

    $$2\times x^2-x-10$$

47. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$2\times x^2-x-10=0$$

48. Calculate the normalized linear coefficient, then halve it.

    $$-\frac{1}{2\times 2}$$

49. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

50. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 1=-1$$

51. Calculate this operation before combining the remaining terms.

    $$1\times 4=4$$

52. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 1}{1\times 4}$$

53. Calculate these two products separately.

    $$-\frac{1}{4}$$

54. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 4}$$

55. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{4}=\frac{1}{4}$$

56. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{4}=-\frac{1}{4}$$

57. Square this half-coefficient and compute the new right-hand side separately.

    $${\left(-\frac{1}{4}\right)}^2$$

58. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{1}{4}\right)}^2=\frac{1}{16}$$

59. Calculate this operation before combining the remaining terms.

    $$10\times 1=10$$

60. Calculate this operation before combining the remaining terms.

    $$1\times 2=2$$

61. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{10\times 1}{1\times 2}$$

62. Calculate these two products separately.

    $$\frac{10}{2}$$

63. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=2$$

64. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{2\times 1}{1\times 2}$$

65. Calculate these two products separately.

    $$\frac{2}{2}$$

66. Show the two divisions explicitly.

    $$\frac{\frac{2}{2}}{\frac{2}{2}}$$

67. Write the reduced fraction.

    $$\frac{1}{1}$$

68. Calculate this operation before combining the remaining terms.

    $$2\times \frac{1}{2}=1$$

69. Calculate this operation before combining the remaining terms.

    $$\frac{1}{1}=1$$

70. Calculate this operation before combining the remaining terms.

    $$5\times 1=5$$

71. Show the two divisions explicitly.

    $$\frac{\frac{10}{2}}{\frac{2}{2}}$$

72. Evaluate the numerator division.

    $$\frac{10}{2}=5$$

73. Evaluate the denominator division.

    $$\frac{2}{2}=1$$

74. Write the reduced fraction.

    $$5\times \frac{1}{1}$$

75. Calculate this operation before combining the remaining terms.

    $$10\times \frac{1}{2}=5$$

76. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{1}{4}\right)}^2=\frac{1}{16}$$

77. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=16$$

78. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$5\times \frac{1}{1}=\frac{5\times 16}{1\times 16}$$

79. Calculate the converted numerator; the denominator is now common.

    $$\frac{80}{16}$$

80. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1}{16}=\frac{1\times 1}{16\times 1}$$

81. Calculate the converted numerator; the denominator is now common.

    $$\frac{1}{16}$$

82. Calculate this operation before combining the remaining terms.

    $$80+1=81$$

83. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{81\times 1}{1\times 16}$$

84. Calculate these two products separately.

    $$\frac{81}{16}$$

85. Calculate this operation before combining the remaining terms.

    $$81\times \frac{1}{16}=\frac{81}{16}$$

86. Keep the common denominator and add the signed numerators.

    $$\frac{80+1}{16}=\frac{81}{16}$$

87. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 16}$$

88. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{16}=\frac{1}{16}$$

89. Calculate this operation before combining the remaining terms.

    $$5+\frac{1}{16}=\frac{81}{16}$$

90. Divide by the coefficient of the squared term and move the constant to the other side.

    $$x^2-\frac{x}{2}=5$$

91. Half the linear coefficient is b/(2a). Add its SQUARE to both sides to keep the equation balanced.

    $$x^2-\frac{x}{2}+{\left(-\frac{1}{4}\right)}^2=5+{\left(-\frac{1}{4}\right)}^2$$

92. The left side is a perfect square. Add the fractions on the right.

    $${\left(x-\frac{1}{4}\right)}^2=\frac{81}{16}$$

93. Take both square-root signs. Each equation gives a possible root.

    $$(x-\frac{1}{4}=\frac{9}{4},x-\frac{1}{4}=-\frac{9}{4})$$

94. Subtract the half-coefficient from both sides to isolate the variable.

    $$(x=\frac{5}{2},x=-2)$$

95. Calculate this root branch separately, preserving the sign.

    $$x=1\times \sqrt{\frac{81}{16}}+\frac{1}{4}$$

96. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{81}{16}}=\frac{9}{4}$$

97. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{9\times 1}{1\times 4}$$

98. Calculate this operation before combining the remaining terms.

    $$9\times \frac{1}{4}=\frac{9}{4}$$

99. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 9}{1\times 4}$$

100. Calculate these two products separately.

     $$\frac{9}{4}$$

101. Calculate this operation before combining the remaining terms.

     $$1\times \frac{9}{4}=\frac{9}{4}$$

102. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=4$$

103. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{9}{4}=\frac{9\times 1}{4\times 1}$$

104. Calculate the converted numerator; the denominator is now common.

     $$\frac{9}{4}$$

105. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{1}{4}=\frac{1\times 1}{4\times 1}$$

106. Calculate the converted numerator; the denominator is now common.

     $$\frac{1}{4}$$

107. Calculate this operation before combining the remaining terms.

     $$9+1=10$$

108. Keep the common denominator and add the signed numerators.

     $$\frac{9+1}{4}=\frac{10}{4}$$

109. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=2$$

110. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{4\times 1}{1\times 2}$$

111. Calculate these two products separately.

     $$\frac{4}{2}$$

112. Show the two divisions explicitly.

     $$\frac{\frac{4}{2}}{\frac{2}{2}}$$

113. Write the reduced fraction.

     $$2\times \frac{1}{1}$$

114. Calculate this operation before combining the remaining terms.

     $$4\times \frac{1}{2}=2$$

115. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{5\times 1}{1\times 2}$$

116. Calculate this operation before combining the remaining terms.

     $$5\times \frac{1}{2}=\frac{5}{2}$$

117. Show the two divisions explicitly.

     $$\frac{\frac{10}{2}}{\frac{4}{2}}$$

118. Evaluate the numerator division.

     $$\frac{10}{2}=5$$

119. Evaluate the denominator division.

     $$\frac{4}{2}=2$$

120. Write the reduced fraction.

     $$\frac{5}{2}$$

121. Calculate this operation before combining the remaining terms.

     $$\frac{9}{4}+\frac{1}{4}=\frac{5}{2}$$

122. Calculate this root branch separately, preserving the sign.

     $$x=-\sqrt{\frac{81}{16}}+\frac{1}{4}$$

123. Calculate this operation before combining the remaining terms.

     $$\sqrt{\frac{81}{16}}=\frac{9}{4}$$

124. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-9\right)\times 1}{1\times 4}$$

125. Calculate this operation before combining the remaining terms.

     $$\left(-9\right)\times \frac{1}{4}=-\frac{9}{4}$$

126. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-1\right)\times 9}{1\times 4}$$

127. Calculate these two products separately.

     $$-\frac{9}{4}$$

128. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{9}{4}=-\frac{9}{4}$$

129. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=4$$

130. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$-\frac{9}{4}=\frac{\left(-9\right)\times 1}{4\times 1}$$

131. Calculate the converted numerator; the denominator is now common.

     $$-\frac{9}{4}$$

132. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{1}{4}=\frac{1\times 1}{4\times 1}$$

133. Calculate the converted numerator; the denominator is now common.

     $$\frac{1}{4}$$

134. Calculate this operation before combining the remaining terms.

     $$-9+1=-8$$

135. Keep the common denominator and add the signed numerators.

     $$\frac{-9+1}{4}=-\frac{8}{4}$$

136. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=4$$

137. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{4\times 1}{1\times 4}$$

138. Calculate these two products separately.

     $$\frac{4}{4}$$

139. Show the two divisions explicitly.

     $$\frac{\frac{4}{4}}{\frac{4}{4}}$$

140. Calculate this operation before combining the remaining terms.

     $$4\times \frac{1}{4}=1$$

141. Calculate this operation before combining the remaining terms.

     $$\left(-2\right)\times 1=-2$$

142. Show the two divisions explicitly.

     $$\frac{-\frac{8}{4}}{\frac{4}{4}}$$

143. Evaluate the numerator division.

     $$-\frac{8}{4}=-2$$

144. Evaluate the denominator division.

     $$\frac{4}{4}=1$$

145. Calculate this operation before combining the remaining terms.

     $$1\times 1=1$$

146. Write the reduced fraction.

     $$-2\times \frac{1}{1}$$

147. Calculate this operation before combining the remaining terms.

     $$-\frac{9}{4}+\frac{1}{4}=-2$$

148. Substitute this root into the original equation before simplifying.

     $$0=0$$

149. Calculate both sides; their equal values confirm the root.

     $$0=0$$

150. Substitute this root into the original equation before simplifying.

     $$0=0$$

151. Calculate both sides; their equal values confirm the root.

     $$0=0$$

152. Method: formula.

153. Substitute the function expressions, keeping the brackets.

     $$-(-2\times x^2+x+10)-0$$

154. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

     $$2\times x^2-x-10$$

155. Write EVERY distributed product before collecting terms.

     $$-10-x+2\times x^2-0$$

156. Calculate this individual product, including its sign and powers.

     $$-10=-10$$

157. Calculate this individual product, including its sign and powers.

     $$-x=-x$$

158. Calculate this individual product, including its sign and powers.

     $$2\times x^2=2\times x^2$$

159. Calculate this individual product, including its sign and powers.

     $$\left(-1\right)\times 0=0$$

160. These terms have the same variable part. Add ONLY their coefficients.

     $$-10+0=-10$$

161. These terms have the same variable part. Add ONLY their coefficients.

     $$-1=-1$$

162. These terms have the same variable part. Add ONLY their coefficients.

     $$2=2$$

163. Put the simplified groups together.

     $$2\times x^2-x-10$$

164. Collect terms with the same power and simplify their coefficients.

     $$2\times x^2-x-10$$

165. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

     $$2\times x^2-x-10=0$$

166. Calculate this operation before combining the remaining terms.

     $$8\times \left(-10\right)=-80$$

167. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \left(-80\right)=80$$

168. Compute the discriminant one operation at a time: square b, multiply 4,a,c, then subtract.

     $${\left(-1\right)}^2-8\times \left(-10\right)$$

169. Calculate this operation before combining the remaining terms.

     $${\left(-1\right)}^2=1$$

170. Calculate this operation before combining the remaining terms.

     $$\left(-4\right)\times 2\times \left(-10\right)=80$$

171. Calculate this operation before combining the remaining terms.

     $$1+80=81$$

172. Compare with ax² + bx + c = 0 and read a, b, c WITH their signs.

     $$(a=2,b=-1,c=-10)$$

173. Compute D = b² − 4ac. A minus times a minus becomes a plus.

     $$D={\left(-1\right)}^2-8\times \left(-10\right)$$

174. Simplify the discriminant.

     $$D=81$$

175. Use x = (−b ± √D)/(2a). Show the plus and minus cases separately.

     $$(x=\frac{1+9}{4},x=\frac{1-9}{4})$$

176. Simplify each numerator and divide by 2a.

     $$\{-2,\frac{5}{2}\}$$

177. Calculate this root branch separately, preserving the sign.

     $$x=\frac{1+1\times \sqrt{81}}{2\times 2}$$

178. Calculate this operation before combining the remaining terms.

     $$\sqrt{81}=9$$

179. Calculate this operation before combining the remaining terms.

     $$1\times 9=9$$

180. Calculate this operation before combining the remaining terms.

     $$1+9=10$$

181. Calculate this operation before combining the remaining terms.

     $$2\times 2=4$$

182. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{10\times 1}{1\times 4}$$

183. Calculate these two products separately.

     $$\frac{10}{4}$$

184. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=2$$

185. Show the two divisions explicitly.

     $$\frac{\frac{10}{2}}{\frac{4}{2}}$$

186. Evaluate the numerator division.

     $$\frac{10}{2}=5$$

187. Evaluate the denominator division.

     $$\frac{4}{2}=2$$

188. Write the reduced fraction.

     $$\frac{5}{2}$$

189. Calculate this operation before combining the remaining terms.

     $$10\times \frac{1}{4}=\frac{5}{2}$$

190. Calculate this root branch separately, preserving the sign.

     $$x=\frac{1-\sqrt{81}}{2\times 2}$$

191. Calculate this operation before combining the remaining terms.

     $$\sqrt{81}=9$$

192. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 9=-9$$

193. Calculate this operation before combining the remaining terms.

     $$1-9=-8$$

194. Calculate this operation before combining the remaining terms.

     $$2\times 2=4$$

195. Calculate this operation before combining the remaining terms.

     $$\left(-8\right)\times 1=-8$$

196. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-8\right)\times 1}{1\times 4}$$

197. Calculate these two products separately.

     $$-\frac{8}{4}$$

198. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=4$$

199. Show the two divisions explicitly.

     $$\frac{-\frac{8}{4}}{\frac{4}{4}}$$

200. Evaluate the numerator division.

     $$-\frac{8}{4}=-2$$

201. Evaluate the denominator division.

     $$\frac{4}{4}=1$$

202. Write the reduced fraction.

     $$-2\times \frac{1}{1}$$

203. Calculate this operation before combining the remaining terms.

     $$\left(-8\right)\times \frac{1}{4}=-2$$

204. Substitute this root into the original equation before simplifying.

     $$0=0$$

205. Calculate both sides; their equal values confirm the root.

     $$0=0$$

206. Substitute this root into the original equation before simplifying.

     $$0=0$$

207. Calculate both sides; their equal values confirm the root.

     $$0=0$$

Answer

The final answer is shown below.