Mathematics 10 — Exercise Review · Q2 (i)

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.

   $$8\times x^2-(x+7)$$

3. Distribute each bracket product: each term multiplies every term in the other bracket.

   $$8\times x^2-x-7$$

4. Write EVERY distributed product before collecting terms.

   $$8\times x^2-7-x$$

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

   $$8\times x^2=8\times x^2$$

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

   $$\left(-1\right)\times 7=-7$$

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

   $$-x=-x$$

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

   $$8=8$$

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

   $$-7=-7$$

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

    $$-1=-1$$

11. Put the simplified groups together.

    $$8\times x^2-7-x$$

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

    $$8\times x^2-x-7$$

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

    $$8\times x^2-x-7=0$$

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

    $$(7-8=-1,7\times \left(-8\right)=-56)$$

15. Rewrite the middle term using those two parts.

    $$8\times x^2+7\times x-8\times x-7=0$$

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

    $$x\times \left(8\times x+7\right)-(8\times x+7)=0$$

17. The common bracket can now be factored out.

    $$\left(x-1\right)\times \left(8\times x+7\right)=0$$

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

    $$(x-1=0,8\times x+7=0)$$

19. Take this factor equal to zero.

    $$x-1=0$$

20. Subtract the constant from both sides.

    $$x-1+1=0+1$$

21. Simplify both sides.

    $$x=1$$

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

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

23. Reduce the fraction to obtain this root.

    $$x=1$$

24. Take this factor equal to zero.

    $$8\times x+7=0$$

25. Subtract the constant from both sides.

    $$8\times x+7-7=0-7$$

26. Simplify both sides.

    $$8\times x=-7$$

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

    $$x=-\frac{7}{8}$$

28. Reduce the fraction to obtain this root.

    $$x=-\frac{7}{8}$$

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

    $$8=8$$

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

    $$8=8$$

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

    $$\frac{49}{8}=\frac{49}{8}$$

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

    $$\frac{49}{8}=\frac{49}{8}$$

33. Method: square.

34. Substitute the function expressions, keeping the brackets.

    $$8\times x^2-(x+7)$$

35. Distribute each bracket product: each term multiplies every term in the other bracket.

    $$8\times x^2-x-7$$

36. Write EVERY distributed product before collecting terms.

    $$8\times x^2-7-x$$

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

    $$8\times x^2=8\times x^2$$

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

    $$\left(-1\right)\times 7=-7$$

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

    $$-x=-x$$

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

    $$8=8$$

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

    $$-7=-7$$

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

    $$-1=-1$$

43. Put the simplified groups together.

    $$8\times x^2-7-x$$

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

    $$8\times x^2-x-7$$

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

    $$8\times x^2-x-7=0$$

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

    $$-\frac{1}{2\times 8}$$

47. Calculate this operation before combining the remaining terms.

    $$2\times 8=16$$

48. Calculate this operation before combining the remaining terms.

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

49. Calculate this operation before combining the remaining terms.

    $$1\times 16=16$$

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

    $$\frac{\left(-1\right)\times 1}{1\times 16}$$

51. Calculate these two products separately.

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

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

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

53. Calculate this operation before combining the remaining terms.

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

54. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{16}=-\frac{1}{16}$$

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

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

56. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{1}{16}\right)}^2=\frac{1}{256}$$

57. Calculate this operation before combining the remaining terms.

    $$7\times 1=7$$

58. Calculate this operation before combining the remaining terms.

    $$1\times 8=8$$

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

    $$\frac{7\times 1}{1\times 8}$$

60. Calculate these two products separately.

    $$\frac{7}{8}$$

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

    $$\frac{1\times 1}{1\times 8}$$

62. Calculate these two products separately.

    $$\frac{1}{8}$$

63. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{8}=\frac{1}{8}$$

64. Calculate this operation before combining the remaining terms.

    $$7\times \frac{1}{8}=\frac{7}{8}$$

65. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{1}{16}\right)}^2=\frac{1}{256}$$

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

    $$\mathrm{LCD}=256$$

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

    $$\frac{7}{8}=\frac{7\times 32}{8\times 32}$$

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

    $$\frac{224}{256}$$

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

    $$\frac{1}{256}=\frac{1\times 1}{256\times 1}$$

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

    $$\frac{1}{256}$$

71. Calculate this operation before combining the remaining terms.

    $$224+1=225$$

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

    $$\frac{225\times 1}{1\times 256}$$

73. Calculate these two products separately.

    $$\frac{225}{256}$$

74. Calculate this operation before combining the remaining terms.

    $$225\times \frac{1}{256}=\frac{225}{256}$$

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

    $$\frac{224+1}{256}=\frac{225}{256}$$

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

    $$\frac{1\times 1}{1\times 256}$$

77. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{256}=\frac{1}{256}$$

78. Calculate this operation before combining the remaining terms.

    $$\frac{7}{8}+\frac{1}{256}=\frac{225}{256}$$

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

    $$x^2-\frac{x}{8}=\frac{7}{8}$$

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

    $$x^2-\frac{x}{8}+{\left(-\frac{1}{16}\right)}^2=\frac{7}{8}+{\left(-\frac{1}{16}\right)}^2$$

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

    $${\left(x-\frac{1}{16}\right)}^2=\frac{225}{256}$$

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

    $$(x-\frac{1}{16}=\frac{15}{16},x-\frac{1}{16}=-\frac{15}{16})$$

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

    $$(x=1,x=-\frac{7}{8})$$

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

    $$x=1\times \sqrt{\frac{225}{256}}+\frac{1}{16}$$

85. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{225}{256}}=\frac{15}{16}$$

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

    $$\frac{15\times 1}{1\times 16}$$

87. Calculate this operation before combining the remaining terms.

    $$15\times \frac{1}{16}=\frac{15}{16}$$

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

    $$\frac{1\times 15}{1\times 16}$$

89. Calculate these two products separately.

    $$\frac{15}{16}$$

90. Calculate this operation before combining the remaining terms.

    $$1\times \frac{15}{16}=\frac{15}{16}$$

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

    $$\mathrm{LCD}=16$$

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

    $$\frac{15}{16}=\frac{15\times 1}{16\times 1}$$

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

    $$\frac{15}{16}$$

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

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

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

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

96. Calculate this operation before combining the remaining terms.

    $$15+1=16$$

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

    $$\frac{15+1}{16}=\frac{16}{16}$$

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

    $$\mathrm{GCD}=16$$

99. Calculate this operation before combining the remaining terms.

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

100. Calculate this operation before combining the remaining terms.

     $$1\times 1=1$$

101. Show the two divisions explicitly.

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

102. Evaluate the numerator division.

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

103. Evaluate the denominator division.

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

104. Write the reduced fraction.

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

105. Calculate this operation before combining the remaining terms.

     $$\frac{15}{16}+\frac{1}{16}=1$$

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

     $$x=-\sqrt{\frac{225}{256}}+\frac{1}{16}$$

107. Calculate this operation before combining the remaining terms.

     $$\sqrt{\frac{225}{256}}=\frac{15}{16}$$

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

     $$\frac{\left(-15\right)\times 1}{1\times 16}$$

109. Calculate this operation before combining the remaining terms.

     $$\left(-15\right)\times \frac{1}{16}=-\frac{15}{16}$$

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

     $$\frac{\left(-1\right)\times 15}{1\times 16}$$

111. Calculate these two products separately.

     $$-\frac{15}{16}$$

112. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{15}{16}=-\frac{15}{16}$$

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

     $$\mathrm{LCD}=16$$

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

     $$-\frac{15}{16}=\frac{\left(-15\right)\times 1}{16\times 1}$$

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

     $$-\frac{15}{16}$$

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

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

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

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

118. Calculate this operation before combining the remaining terms.

     $$-15+1=-14$$

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

     $$\frac{-15+1}{16}=-\frac{14}{16}$$

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

     $$\mathrm{GCD}=2$$

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

     $$\frac{\left(-14\right)\times 1}{1\times 2}$$

122. Calculate these two products separately.

     $$-\frac{14}{2}$$

123. Show the two divisions explicitly.

     $$\frac{-\frac{14}{2}}{\frac{2}{2}}$$

124. Evaluate the denominator division.

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

125. Write the reduced fraction.

     $$-7\times \frac{1}{1}$$

126. Calculate this operation before combining the remaining terms.

     $$\left(-14\right)\times \frac{1}{2}=-7$$

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

     $$\frac{16\times 1}{1\times 2}$$

128. Calculate these two products separately.

     $$\frac{16}{2}$$

129. Show the two divisions explicitly.

     $$\frac{\frac{16}{2}}{\frac{2}{2}}$$

130. Write the reduced fraction.

     $$8\times \frac{1}{1}$$

131. Calculate this operation before combining the remaining terms.

     $$16\times \frac{1}{2}=8$$

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

     $$\frac{\left(-7\right)\times 1}{1\times 8}$$

133. Calculate this operation before combining the remaining terms.

     $$\left(-7\right)\times \frac{1}{8}=-\frac{7}{8}$$

134. Show the two divisions explicitly.

     $$\frac{-\frac{14}{2}}{\frac{16}{2}}$$

135. Evaluate the numerator division.

     $$-\frac{14}{2}=-7$$

136. Evaluate the denominator division.

     $$\frac{16}{2}=8$$

137. Write the reduced fraction.

     $$-\frac{7}{8}$$

138. Calculate this operation before combining the remaining terms.

     $$-\frac{15}{16}+\frac{1}{16}=-\frac{7}{8}$$

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

     $$8=8$$

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

     $$8=8$$

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

     $$\frac{49}{8}=\frac{49}{8}$$

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

     $$\frac{49}{8}=\frac{49}{8}$$

143. Method: formula.

144. Substitute the function expressions, keeping the brackets.

     $$8\times x^2-(x+7)$$

145. Distribute each bracket product: each term multiplies every term in the other bracket.

     $$8\times x^2-x-7$$

146. Write EVERY distributed product before collecting terms.

     $$8\times x^2-7-x$$

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

     $$8\times x^2=8\times x^2$$

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

     $$\left(-1\right)\times 7=-7$$

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

     $$-x=-x$$

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

     $$8=8$$

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

     $$-7=-7$$

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

     $$-1=-1$$

153. Put the simplified groups together.

     $$8\times x^2-7-x$$

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

     $$8\times x^2-x-7$$

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

     $$8\times x^2-x-7=0$$

156. Calculate this operation before combining the remaining terms.

     $$32\times \left(-7\right)=-224$$

157. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \left(-224\right)=224$$

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

     $${\left(-1\right)}^2-32\times \left(-7\right)$$

159. Calculate this operation before combining the remaining terms.

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

160. Calculate this operation before combining the remaining terms.

     $$\left(-4\right)\times 8\times \left(-7\right)=224$$

161. Calculate this operation before combining the remaining terms.

     $$1+224=225$$

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

     $$(a=8,b=-1,c=-7)$$

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

     $$D={\left(-1\right)}^2-32\times \left(-7\right)$$

164. Simplify the discriminant.

     $$D=225$$

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

     $$(x=\frac{1+15}{16},x=\frac{1-15}{16})$$

166. Simplify each numerator and divide by 2a.

     $$\{-\frac{7}{8},1\}$$

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

     $$x=\frac{1+1\times \sqrt{225}}{2\times 8}$$

168. Calculate this operation before combining the remaining terms.

     $$\sqrt{225}=15$$

169. Calculate this operation before combining the remaining terms.

     $$1\times 15=15$$

170. Calculate this operation before combining the remaining terms.

     $$1+15=16$$

171. Calculate this operation before combining the remaining terms.

     $$2\times 8=16$$

172. Calculate this operation before combining the remaining terms.

     $$16\times 1=16$$

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

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

174. Calculate these two products separately.

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

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

     $$\mathrm{GCD}=16$$

176. Show the two divisions explicitly.

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

177. Evaluate the numerator division.

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

178. Evaluate the denominator division.

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

179. Write the reduced fraction.

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

180. Calculate this operation before combining the remaining terms.

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

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

     $$x=\frac{1-\sqrt{225}}{2\times 8}$$

182. Calculate this operation before combining the remaining terms.

     $$\sqrt{225}=15$$

183. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 15=-15$$

184. Calculate this operation before combining the remaining terms.

     $$1-15=-14$$

185. Calculate this operation before combining the remaining terms.

     $$2\times 8=16$$

186. Calculate this operation before combining the remaining terms.

     $$\left(-14\right)\times 1=-14$$

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

     $$\frac{\left(-14\right)\times 1}{1\times 16}$$

188. Calculate these two products separately.

     $$-\frac{14}{16}$$

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

     $$\mathrm{GCD}=2$$

190. Show the two divisions explicitly.

     $$\frac{-\frac{14}{2}}{\frac{16}{2}}$$

191. Evaluate the numerator division.

     $$-\frac{14}{2}=-7$$

192. Evaluate the denominator division.

     $$\frac{16}{2}=8$$

193. Write the reduced fraction.

     $$-\frac{7}{8}$$

194. Calculate this operation before combining the remaining terms.

     $$\left(-14\right)\times \frac{1}{16}=-\frac{7}{8}$$

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

     $$8=8$$

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

     $$8=8$$

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

     $$\frac{49}{8}=\frac{49}{8}$$

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

     $$\frac{49}{8}=\frac{49}{8}$$

Answer

The final answer is shown below.