Mathematics 10 — Exercise 2.1 · Q4 (v)

Full question

Use the quadratic formula to solve the following equation: (x+4)/(x-4)+(x-2)/(x-3) = 6⅓. where x ≠ 4, 3.

Step-by-step solution

1. A denominator cannot equal zero. Keep the original excluded values throughout the calculation.

   $$3\times x^2-21\times x+36\ne 0$$

2. Multiply BOTH sides by the common denominator to remove the fractions.

   $$\left(\frac{x-2}{x-3}+\frac{x+4}{x-4}\right)\times \left(3\times x^2-21\times x+36\right)=\frac{57\times x^2-399\times x+684}{3}$$

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

   $$13\times x^2-118\times x+240=0$$

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

   $${\left(-118\right)}^2-52\times 240$$

5. Calculate this operation before combining the remaining terms.

   $${\left(-118\right)}^2=13924$$

6. Calculate this operation before combining the remaining terms.

   $$\left(-4\right)\times 13\times 240=-12480$$

7. Calculate this operation before combining the remaining terms.

   $$13924-12480=1444$$

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

   $$(a=13,b=-118,c=240)$$

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

   $$D={\left(-118\right)}^2-52\times 240$$

10. Simplify the discriminant.

    $$D=1444$$

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

    $$(x=\frac{118+38}{26},x=\frac{118-38}{26})$$

12. Simplify each numerator and divide by 2a.

    $$\{\frac{40}{13},6\}$$

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

    $$x=\frac{1\times \sqrt{1444}+118}{2\times 13}$$

14. Calculate this operation before combining the remaining terms.

    $$\sqrt{1444}=38$$

15. Calculate this operation before combining the remaining terms.

    $$1\times 38=38$$

16. Calculate this operation before combining the remaining terms.

    $$118+38=156$$

17. Calculate this operation before combining the remaining terms.

    $$2\times 13=26$$

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

    $$\frac{156\times 1}{1\times 26}$$

19. Calculate these two products separately.

    $$\frac{156}{26}$$

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

    $$\mathrm{GCD}=26$$

21. Show the two divisions explicitly.

    $$\frac{\frac{156}{26}}{\frac{26}{26}}$$

22. Evaluate the numerator division.

    $$\frac{156}{26}=6$$

23. Evaluate the denominator division.

    $$\frac{26}{26}=1$$

24. Write the reduced fraction.

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

25. Calculate this operation before combining the remaining terms.

    $$156\times \frac{1}{26}=6$$

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

    $$x=\frac{118-\sqrt{1444}}{2\times 13}$$

27. Calculate this operation before combining the remaining terms.

    $$\sqrt{1444}=38$$

28. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 38=-38$$

29. Calculate this operation before combining the remaining terms.

    $$118-38=80$$

30. Calculate this operation before combining the remaining terms.

    $$2\times 13=26$$

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

    $$\frac{80\times 1}{1\times 26}$$

32. Calculate these two products separately.

    $$\frac{80}{26}$$

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

    $$\mathrm{GCD}=2$$

34. Show the two divisions explicitly.

    $$\frac{\frac{80}{2}}{\frac{26}{2}}$$

35. Evaluate the numerator division.

    $$\frac{80}{2}=40$$

36. Evaluate the denominator division.

    $$\frac{26}{2}=13$$

37. Write the reduced fraction.

    $$\frac{40}{13}$$

38. Calculate this operation before combining the remaining terms.

    $$80\times \frac{1}{26}=\frac{40}{13}$$

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

    $$\frac{19}{3}=\frac{19}{3}$$

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

    $$\frac{19}{3}=\frac{19}{3}$$

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

    $$\frac{19}{3}=\frac{19}{3}$$

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

    $$\frac{19}{3}=\frac{19}{3}$$

Answer

The final answer is shown below.