Mathematics 10 — Exercise 2.1 · Q3 (i)

Full question

Solve the following quadratic equation by completing square method: 2 × x²+5 × x+2 = 0.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

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

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

   $$2\times x^2+5\times x+2$$

3. Write EVERY distributed product before collecting terms.

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

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

   $$2=2$$

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

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

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

   $$5\times x=5\times x$$

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

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

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

   $$2+0=2$$

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

   $$2=2$$

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

    $$5=5$$

11. Put the simplified groups together.

    $$2\times x^2+5\times x+2$$

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

    $$2\times x^2+5\times x+2$$

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

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

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

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

15. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

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

    $$\frac{5\times 1}{1\times 4}$$

17. Calculate these two products separately.

    $$\frac{5}{4}$$

18. Calculate this operation before combining the remaining terms.

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

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

    $${\left(\frac{5}{4}\right)}^2$$

20. Calculate this operation before combining the remaining terms.

    $${\left(\frac{5}{4}\right)}^2=\frac{25}{16}$$

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

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

22. Calculate these two products separately.

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

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

    $$\mathrm{GCD}=2$$

24. Show the two divisions explicitly.

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

25. Evaluate the numerator division.

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

26. Evaluate the denominator division.

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

27. Write the reduced fraction.

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

28. Calculate this operation before combining the remaining terms.

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

29. Calculate this operation before combining the remaining terms.

    $${\left(\frac{5}{4}\right)}^2=\frac{25}{16}$$

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

    $$\mathrm{LCD}=16$$

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

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

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

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

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

    $$\frac{25}{16}=\frac{25\times 1}{16\times 1}$$

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

    $$\frac{25}{16}$$

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

    $$\frac{-16+25}{16}=\frac{9}{16}$$

36. Calculate this operation before combining the remaining terms.

    $$-1+\frac{25}{16}=\frac{9}{16}$$

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

    $$x^2+\frac{5\times x}{2}=-1$$

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

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

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

    $${\left(x+\frac{5}{4}\right)}^2=\frac{9}{16}$$

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

    $$(x+\frac{5}{4}=\frac{3}{4},x+\frac{5}{4}=-\frac{3}{4})$$

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

    $$(x=-\frac{1}{2},x=-2)$$

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

    $$x=1\times \sqrt{\frac{9}{16}}-\frac{5}{4}$$

43. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{9}{16}}=\frac{3}{4}$$

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

    $$\frac{1\times 3}{1\times 4}$$

45. Calculate these two products separately.

    $$\frac{3}{4}$$

46. Calculate this operation before combining the remaining terms.

    $$1\times \frac{3}{4}=\frac{3}{4}$$

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

    $$\mathrm{LCD}=4$$

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

    $$\frac{3}{4}=\frac{3\times 1}{4\times 1}$$

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

    $$\frac{3}{4}$$

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

    $$-\frac{5}{4}=\frac{\left(-5\right)\times 1}{4\times 1}$$

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

    $$-\frac{5}{4}$$

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

    $$\frac{3-5}{4}=-\frac{2}{4}$$

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

    $$\mathrm{GCD}=2$$

54. Show the two divisions explicitly.

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

55. Evaluate the numerator division.

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

56. Evaluate the denominator division.

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

57. Write the reduced fraction.

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

58. Calculate this operation before combining the remaining terms.

    $$\frac{3}{4}-\frac{5}{4}=-\frac{1}{2}$$

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

    $$x=-\sqrt{\frac{9}{16}}-\frac{5}{4}$$

60. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{9}{16}}=\frac{3}{4}$$

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

    $$\frac{\left(-1\right)\times 3}{1\times 4}$$

62. Calculate these two products separately.

    $$-\frac{3}{4}$$

63. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{3}{4}=-\frac{3}{4}$$

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

    $$\mathrm{LCD}=4$$

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

    $$-\frac{3}{4}=\frac{\left(-3\right)\times 1}{4\times 1}$$

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

    $$-\frac{3}{4}$$

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

    $$-\frac{5}{4}=\frac{\left(-5\right)\times 1}{4\times 1}$$

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

    $$-\frac{5}{4}$$

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

    $$\frac{-3-5}{4}=-\frac{8}{4}$$

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

    $$\mathrm{GCD}=4$$

71. Show the two divisions explicitly.

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

72. Evaluate the numerator division.

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

73. Evaluate the denominator division.

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

74. Write the reduced fraction.

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

75. Calculate this operation before combining the remaining terms.

    $$-\frac{3}{4}-\frac{5}{4}=-2$$

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

    $$0=0$$

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

    $$0=0$$

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

    $$0=0$$

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

    $$0=0$$

Answer

The final answer is shown below.