Mathematics 10 — Exercise 2.1 · Q2 (vi)

Full question

Solve the following quadratic equation by factorization method: 1/(x-1)-1/(x+3) = 1/35. where x ≠ 1, −3.

Step-by-step solution

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

   $$35\times x^2+70\times x-105\ne 0$$

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

   $$\left(-\frac{1}{x+3}+\frac{1}{x-1}\right)\times \left(35\times x^2+70\times x-105\right)=\frac{35\times x^2+70\times x-105}{35}$$

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

   $$x^2+2\times x-143=0$$

4. Split the middle coefficient into two numbers whose sum is b and whose product is a·c.

   $$(13-11=2,13\times \left(-11\right)=-143)$$

5. Rewrite the middle term using those two parts.

   $$x^2+13\times x-11\times x-143=0$$

6. Group the first two terms and the last two. Take out the common factor from each group.

   $$x\times \left(x+13\right)-11\times \left(x+13\right)=0$$

7. The common bracket can now be factored out.

   $$\left(x-11\right)\times \left(x+13\right)=0$$

8. A product is zero only if at least one factor is zero. Set each linear factor equal to zero.

   $$(x-11=0,x+13=0)$$

9. Take this factor equal to zero.

   $$x-11=0$$

10. Subtract the constant from both sides.

    $$x-11+11=0+11$$

11. Simplify both sides.

    $$x=11$$

12. Divide both sides by the coefficient of the variable.

    $$x=11\times \frac{1}{1}$$

13. Reduce the fraction to obtain this root.

    $$x=11$$

14. Take this factor equal to zero.

    $$x+13=0$$

15. Subtract the constant from both sides.

    $$x+13-13=0-13$$

16. Simplify both sides.

    $$x=-13$$

17. Divide both sides by the coefficient of the variable.

    $$x=-13\times \frac{1}{1}$$

18. Reduce the fraction to obtain this root.

    $$x=-13$$

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

    $$\frac{1}{35}=\frac{1}{35}$$

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

    $$\frac{1}{35}=\frac{1}{35}$$

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

    $$\frac{1}{35}=\frac{1}{35}$$

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

    $$\frac{1}{35}=\frac{1}{35}$$

Answer

The final answer is shown below.