Mathematics 10 — Exercise 2.1 · Q4 (iii)

Full question

Use the quadratic formula to solve the following equation: 2 × x²+7 × x = 15.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$2\times x^2+7\times x-15$$

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

   $$2\times x^2+7\times x-15$$

3. Write EVERY distributed product before collecting terms.

   $$2\times x^2+7\times x-15$$

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

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

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

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

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

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

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

   $$2=2$$

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

   $$7=7$$

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

   $$-15=-15$$

10. Put the simplified groups together.

    $$2\times x^2+7\times x-15$$

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

    $$2\times x^2+7\times x-15$$

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

    $$2\times x^2+7\times x-15=0$$

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

    $$7^2-8\times \left(-15\right)$$

14. Calculate this operation before combining the remaining terms.

    $$7^2=49$$

15. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 2\times \left(-15\right)=120$$

16. Calculate this operation before combining the remaining terms.

    $$49+120=169$$

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

    $$(a=2,b=7,c=-15)$$

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

    $$D=7^2-8\times \left(-15\right)$$

19. Simplify the discriminant.

    $$D=169$$

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

    $$(x=\frac{-7+13}{4},x=\frac{-7-13}{4})$$

21. Simplify each numerator and divide by 2a.

    $$\{-5,\frac{3}{2}\}$$

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

    $$x=\frac{-7+1\times \sqrt{169}}{2\times 2}$$

23. Calculate this operation before combining the remaining terms.

    $$\sqrt{169}=13$$

24. Calculate this operation before combining the remaining terms.

    $$1\times 13=13$$

25. Calculate this operation before combining the remaining terms.

    $$-7+13=6$$

26. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

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

    $$\frac{6\times 1}{1\times 4}$$

28. Calculate these two products separately.

    $$\frac{6}{4}$$

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

    $$\mathrm{GCD}=2$$

30. Show the two divisions explicitly.

    $$\frac{\frac{6}{2}}{\frac{4}{2}}$$

31. Evaluate the numerator division.

    $$\frac{6}{2}=3$$

32. Evaluate the denominator division.

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

33. Write the reduced fraction.

    $$\frac{3}{2}$$

34. Calculate this operation before combining the remaining terms.

    $$6\times \frac{1}{4}=\frac{3}{2}$$

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

    $$x=\frac{\left(-1\right)\times \sqrt{169}-7}{2\times 2}$$

36. Calculate this operation before combining the remaining terms.

    $$\sqrt{169}=13$$

37. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 13=-13$$

38. Calculate this operation before combining the remaining terms.

    $$-7-13=-20$$

39. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

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

    $$\frac{\left(-20\right)\times 1}{1\times 4}$$

41. Calculate these two products separately.

    $$-\frac{20}{4}$$

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

    $$\mathrm{GCD}=4$$

43. Show the two divisions explicitly.

    $$\frac{-\frac{20}{4}}{\frac{4}{4}}$$

44. Evaluate the numerator division.

    $$-\frac{20}{4}=-5$$

45. Evaluate the denominator division.

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

46. Write the reduced fraction.

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

47. Calculate this operation before combining the remaining terms.

    $$\left(-20\right)\times \frac{1}{4}=-5$$

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

    $$15=15$$

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

    $$15=15$$

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

    $$15=15$$

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

    $$15=15$$

Answer

The final answer is shown below.