Mathematics 10 — Exercise 2.1 · Q3 (ii)

Full question

Solve the following quadratic equation by completing square method: x²+x = 42.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$x^2+x-42$$

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

   $$x^2+x-42$$

3. Write EVERY distributed product before collecting terms.

   $$x+x^2-42$$

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

   $$x=x$$

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

   $$x^2=x^2$$

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

   $$\left(-1\right)\times 42=-42$$

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

   $$1=1$$

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

   $$1=1$$

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

   $$-42=-42$$

10. Put the simplified groups together.

    $$x+x^2-42$$

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

    $$x^2+x-42$$

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

    $$x^2+x-42=0$$

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

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

14. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

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

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

16. Calculate these two products separately.

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

17. Calculate this operation before combining the remaining terms.

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

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

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

19. Calculate this operation before combining the remaining terms.

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

20. Calculate this operation before combining the remaining terms.

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

21. Calculate this operation before combining the remaining terms.

    $$42\times 1=42$$

22. Calculate this operation before combining the remaining terms.

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

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

    $$\mathrm{LCD}=4$$

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

    $$42\times \frac{1}{1}=\frac{42\times 4}{1\times 4}$$

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

    $$\frac{168}{4}$$

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

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

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

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

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

    $$\frac{168+1}{4}=\frac{169}{4}$$

29. Calculate this operation before combining the remaining terms.

    $$42+\frac{1}{4}=\frac{169}{4}$$

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

    $$x^2+x=42$$

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

    $$x^2+x+{\left(\frac{1}{2}\right)}^2=42+{\left(\frac{1}{2}\right)}^2$$

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

    $${\left(x+\frac{1}{2}\right)}^2=\frac{169}{4}$$

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

    $$(x+\frac{1}{2}=\frac{13}{2},x+\frac{1}{2}=-\frac{13}{2})$$

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

    $$(x=6,x=-7)$$

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

    $$x=1\times \sqrt{\frac{169}{4}}-\frac{1}{2}$$

36. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{169}{4}}=\frac{13}{2}$$

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

    $$\frac{1\times 13}{1\times 2}$$

38. Calculate these two products separately.

    $$\frac{13}{2}$$

39. Calculate this operation before combining the remaining terms.

    $$1\times \frac{13}{2}=\frac{13}{2}$$

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

    $$\mathrm{LCD}=2$$

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

    $$\frac{13}{2}=\frac{13\times 1}{2\times 1}$$

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

    $$\frac{13}{2}$$

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

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

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

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

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

    $$\frac{13-1}{2}=\frac{12}{2}$$

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

    $$\mathrm{GCD}=2$$

47. Show the two divisions explicitly.

    $$\frac{\frac{12}{2}}{\frac{2}{2}}$$

48. Evaluate the numerator division.

    $$\frac{12}{2}=6$$

49. Evaluate the denominator division.

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

50. Write the reduced fraction.

    $$6\times \frac{1}{1}$$

51. Calculate this operation before combining the remaining terms.

    $$\frac{13}{2}-\frac{1}{2}=6$$

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

    $$x=-\sqrt{\frac{169}{4}}-\frac{1}{2}$$

53. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{169}{4}}=\frac{13}{2}$$

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

    $$\frac{\left(-1\right)\times 13}{1\times 2}$$

55. Calculate these two products separately.

    $$-\frac{13}{2}$$

56. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{13}{2}=-\frac{13}{2}$$

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

    $$\mathrm{LCD}=2$$

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

    $$-\frac{13}{2}=\frac{\left(-13\right)\times 1}{2\times 1}$$

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

    $$-\frac{13}{2}$$

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

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

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

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

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

    $$\frac{-13-1}{2}=-\frac{14}{2}$$

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

    $$\mathrm{GCD}=2$$

64. Show the two divisions explicitly.

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

65. Evaluate the numerator division.

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

66. Evaluate the denominator division.

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

67. Write the reduced fraction.

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

68. Calculate this operation before combining the remaining terms.

    $$-\frac{13}{2}-\frac{1}{2}=-7$$

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

    $$42=42$$

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

    $$42=42$$

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

    $$42=42$$

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

    $$42=42$$

Answer

The final answer is shown below.