Mathematics 10 — Exercise 2.1 · Q2 (v)

Full question

Solve the following quadratic equation by factorization method: (3 × x-8)/(x-2) = (5 × x-2)/(x+5). where x ≠ 2, −5.

Step-by-step solution

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

   $$x^2+3\times x-10\ne 0$$

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

   $$\left(\frac{3\times x-8}{x-2}\right)\times \left(x^2+3\times x-10\right)=\left(\frac{5\times x-2}{x+5}\right)\times \left(x^2+3\times x-10\right)$$

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

   $$2\times x^2-19\times x+44=0$$

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

   $$(-11-8=-19,\left(-11\right)\times \left(-8\right)=88)$$

5. Rewrite the middle term using those two parts.

   $$2\times x^2-11\times x-8\times x+44=0$$

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

   $$x\times \left(2\times x-11\right)-4\times \left(2\times x-11\right)=0$$

7. The common bracket can now be factored out.

   $$\left(x-4\right)\times \left(2\times x-11\right)=0$$

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

   $$(x-4=0,2\times x-11=0)$$

9. Take this factor equal to zero.

   $$x-4=0$$

10. Subtract the constant from both sides.

    $$x-4+4=0+4$$

11. Simplify both sides.

    $$x=4$$

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

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

13. Reduce the fraction to obtain this root.

    $$x=4$$

14. Take this factor equal to zero.

    $$2\times x-11=0$$

15. Subtract the constant from both sides.

    $$2\times x-11+11=0+11$$

16. Simplify both sides.

    $$2\times x=11$$

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

    $$x=\frac{11}{2}$$

18. Reduce the fraction to obtain this root.

    $$x=\frac{11}{2}$$

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

    $$\frac{17}{7}=\frac{17}{7}$$

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

    $$\frac{17}{7}=\frac{17}{7}$$

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

    $$2=2$$

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

    $$2=2$$

Answer

The final answer is shown below.