Mathematics 10 — Exercise 2.1 · Q2 (iv)

Full question

Solve the following quadratic equation by factorization method: x²-3 × x/2 = 9/2.

Step-by-step solution

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

   $$2\ne 0$$

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

   $$2\times \left(x^2-\frac{3\times x}{2}\right)=2\times \frac{9}{2}$$

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

   $$2\times x^2-3\times x-9=0$$

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

   $$(3-6=-3,3\times \left(-6\right)=-18)$$

5. Rewrite the middle term using those two parts.

   $$2\times x^2+3\times x-6\times x-9=0$$

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

   $$x\times \left(2\times x+3\right)-3\times \left(2\times x+3\right)=0$$

7. The common bracket can now be factored out.

   $$\left(x-3\right)\times \left(2\times x+3\right)=0$$

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

   $$(x-3=0,2\times x+3=0)$$

9. Take this factor equal to zero.

   $$x-3=0$$

10. Subtract the constant from both sides.

    $$x-3+3=0+3$$

11. Simplify both sides.

    $$x=3$$

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

    $$x=3\times \frac{1}{1}$$

13. Reduce the fraction to obtain this root.

    $$x=3$$

14. Take this factor equal to zero.

    $$2\times x+3=0$$

15. Subtract the constant from both sides.

    $$2\times x+3-3=0-3$$

16. Simplify both sides.

    $$2\times x=-3$$

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

    $$x=-\frac{3}{2}$$

18. Reduce the fraction to obtain this root.

    $$x=-\frac{3}{2}$$

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

    $$\frac{9}{2}=\frac{9}{2}$$

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

    $$\frac{9}{2}=\frac{9}{2}$$

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

    $$\frac{9}{2}=\frac{9}{2}$$

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

    $$\frac{9}{2}=\frac{9}{2}$$

Answer

The final answer is shown below.