Mathematics 10 — Exercise 2.1 · Q3 (iii)

Full question

Solve the following quadratic equation by completing square method: 12 × x²+7 × x = 12.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$12\times x^2+7\times x-12$$

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

   $$12\times x^2+7\times x-12$$

3. Write EVERY distributed product before collecting terms.

   $$7\times x+12\times x^2-12$$

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

   $$7\times x=7\times x$$

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

   $$12\times x^2=12\times x^2$$

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

   $$\left(-1\right)\times 12=-12$$

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

   $$7=7$$

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

   $$12=12$$

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

   $$-12=-12$$

10. Put the simplified groups together.

    $$7\times x+12\times x^2-12$$

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

    $$12\times x^2+7\times x-12$$

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

    $$12\times x^2+7\times x-12=0$$

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

    $$\frac{7}{2\times 12}$$

14. Calculate this operation before combining the remaining terms.

    $$2\times 12=24$$

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

    $$\frac{7\times 1}{1\times 24}$$

16. Calculate these two products separately.

    $$\frac{7}{24}$$

17. Calculate this operation before combining the remaining terms.

    $$7\times \frac{1}{24}=\frac{7}{24}$$

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

    $${\left(\frac{7}{24}\right)}^2$$

19. Calculate this operation before combining the remaining terms.

    $${\left(\frac{7}{24}\right)}^2=\frac{49}{576}$$

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

    $$\frac{12\times 1}{1\times 12}$$

21. Calculate these two products separately.

    $$\frac{12}{12}$$

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

    $$\mathrm{GCD}=12$$

23. Show the two divisions explicitly.

    $$\frac{\frac{12}{12}}{\frac{12}{12}}$$

24. Evaluate the numerator division.

    $$\frac{12}{12}=1$$

25. Evaluate the denominator division.

    $$\frac{12}{12}=1$$

26. Write the reduced fraction.

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

27. Calculate this operation before combining the remaining terms.

    $$12\times \frac{1}{12}=1$$

28. Calculate this operation before combining the remaining terms.

    $${\left(\frac{7}{24}\right)}^2=\frac{49}{576}$$

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

    $$\mathrm{LCD}=576$$

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

    $$\frac{1}{1}=\frac{1\times 576}{1\times 576}$$

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

    $$\frac{576}{576}$$

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

    $$\frac{49}{576}=\frac{49\times 1}{576\times 1}$$

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

    $$\frac{49}{576}$$

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

    $$\frac{576+49}{576}=\frac{625}{576}$$

35. Calculate this operation before combining the remaining terms.

    $$1+\frac{49}{576}=\frac{625}{576}$$

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

    $$x^2+\frac{7\times x}{12}=1$$

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

    $$x^2+\frac{7\times x}{12}+{\left(\frac{7}{24}\right)}^2=1+{\left(\frac{7}{24}\right)}^2$$

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

    $${\left(x+\frac{7}{24}\right)}^2=\frac{625}{576}$$

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

    $$(x+\frac{7}{24}=\frac{25}{24},x+\frac{7}{24}=-\frac{25}{24})$$

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

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

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

    $$x=1\times \sqrt{\frac{625}{576}}-\frac{7}{24}$$

42. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{625}{576}}=\frac{25}{24}$$

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

    $$\frac{1\times 25}{1\times 24}$$

44. Calculate these two products separately.

    $$\frac{25}{24}$$

45. Calculate this operation before combining the remaining terms.

    $$1\times \frac{25}{24}=\frac{25}{24}$$

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

    $$\mathrm{LCD}=24$$

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

    $$\frac{25}{24}=\frac{25\times 1}{24\times 1}$$

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

    $$\frac{25}{24}$$

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

    $$-\frac{7}{24}=\frac{\left(-7\right)\times 1}{24\times 1}$$

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

    $$-\frac{7}{24}$$

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

    $$\frac{25-7}{24}=\frac{18}{24}$$

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

    $$\mathrm{GCD}=6$$

53. Show the two divisions explicitly.

    $$\frac{\frac{18}{6}}{\frac{24}{6}}$$

54. Evaluate the numerator division.

    $$\frac{18}{6}=3$$

55. Evaluate the denominator division.

    $$\frac{24}{6}=4$$

56. Write the reduced fraction.

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

57. Calculate this operation before combining the remaining terms.

    $$\frac{25}{24}-\frac{7}{24}=\frac{3}{4}$$

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

    $$x=-\sqrt{\frac{625}{576}}-\frac{7}{24}$$

59. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{625}{576}}=\frac{25}{24}$$

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

    $$\frac{\left(-1\right)\times 25}{1\times 24}$$

61. Calculate these two products separately.

    $$-\frac{25}{24}$$

62. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{25}{24}=-\frac{25}{24}$$

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

    $$\mathrm{LCD}=24$$

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

    $$-\frac{25}{24}=\frac{\left(-25\right)\times 1}{24\times 1}$$

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

    $$-\frac{25}{24}$$

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

    $$-\frac{7}{24}=\frac{\left(-7\right)\times 1}{24\times 1}$$

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

    $$-\frac{7}{24}$$

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

    $$\frac{-25-7}{24}=-\frac{32}{24}$$

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

    $$\mathrm{GCD}=8$$

70. Show the two divisions explicitly.

    $$\frac{-\frac{32}{8}}{\frac{24}{8}}$$

71. Evaluate the numerator division.

    $$-\frac{32}{8}=-4$$

72. Evaluate the denominator division.

    $$\frac{24}{8}=3$$

73. Write the reduced fraction.

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

74. Calculate this operation before combining the remaining terms.

    $$-\frac{25}{24}-\frac{7}{24}=-\frac{4}{3}$$

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

    $$12=12$$

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

    $$12=12$$

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

    $$12=12$$

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

    $$12=12$$

Answer

The final answer is shown below.