Mathematics 10 — Exercise 2.1 · Q4 (vi)

Full question

Use the quadratic formula to solve the following equation: (3 × x-3)/(x+1) = (2 × x-1)/(x-1). where x ≠ −1, 1.

Step-by-step solution

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

   $$x^2-1\ne 0$$

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

   $$\left(\frac{3\times x-3}{x+1}\right)\times \left(x^2-1\right)=\left(\frac{2\times x-1}{x-1}\right)\times \left(x^2-1\right)$$

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

   $$x^2-7\times x+4=0$$

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

   $${\left(-7\right)}^2-4\times 4$$

5. Calculate this operation before combining the remaining terms.

   $${\left(-7\right)}^2=49$$

6. Calculate this operation before combining the remaining terms.

   $$\left(-4\right)\times 1\times 4=-16$$

7. Calculate this operation before combining the remaining terms.

   $$49-16=33$$

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

   $$(a=1,b=-7,c=4)$$

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

   $$D={\left(-7\right)}^2-4\times 4$$

10. Simplify the discriminant.

    $$D=33$$

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

    $$(x=\frac{\sqrt{33}+7}{2},x=\frac{7-\sqrt{33}}{2})$$

12. Simplify each numerator and divide by 2a.

    $$\{\frac{7}{2}-\frac{1}{2}\times \sqrt{33},\frac{1}{2}\times \sqrt{33}+\frac{7}{2}\}$$

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

    $$x=\frac{1\times \sqrt{33}+7}{2\times 1}$$

14. Calculate this operation before combining the remaining terms.

    $$1\times \sqrt{33}=\sqrt{33}$$

15. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

16. Calculate this operation before combining the remaining terms.

    $$\sqrt{33}+7\times \frac{1}{2}=\frac{1}{2}\times \sqrt{33}+\frac{7}{2}$$

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

    $$x=\frac{7-\sqrt{33}}{2\times 1}$$

18. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

19. Calculate this operation before combining the remaining terms.

    $$7-\sqrt{33}\times \frac{1}{2}=\frac{7}{2}-\frac{1}{2}\times \sqrt{33}$$

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

    $$\frac{\frac{15}{2}+\frac{3}{2}\times \sqrt{33}}{\frac{1}{2}\times \sqrt{33}+\frac{9}{2}}=\frac{\sqrt{33}+6}{\frac{5}{2}+\frac{1}{2}\times \sqrt{33}}$$

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

    $$\frac{3}{4}+\frac{1}{4}\times \sqrt{33}=\frac{3}{4}+\frac{1}{4}\times \sqrt{33}$$

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

    $$\frac{\frac{15}{2}-\frac{3}{2}\times \sqrt{33}}{\frac{9}{2}-\frac{1}{2}\times \sqrt{33}}=\frac{6-\sqrt{33}}{\frac{5}{2}-\frac{1}{2}\times \sqrt{33}}$$

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

    $$\frac{3}{4}-\frac{1}{4}\times \sqrt{33}=\frac{3}{4}-\frac{1}{4}\times \sqrt{33}$$

Answer

The final answer is shown below.