Mathematics 10 — Exercise 2.1 · Q2 (i)

Full question

Solve the following quadratic equation by factorization method: x²-x-6 = 0.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$--x^2+x+6-0$$

2. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$x^2-x-6$$

3. Write EVERY distributed product before collecting terms.

   $$-6+x^2-x-0$$

4. Calculate this individual product, including its sign and powers.

   $$-6=-6$$

5. Calculate this individual product, including its sign and powers.

   $$x^2=x^2$$

6. Calculate this individual product, including its sign and powers.

   $$-x=-x$$

7. Calculate this individual product, including its sign and powers.

   $$\left(-1\right)\times 0=0$$

8. These terms have the same variable part. Add ONLY their coefficients.

   $$-6+0=-6$$

9. These terms have the same variable part. Add ONLY their coefficients.

   $$1=1$$

10. These terms have the same variable part. Add ONLY their coefficients.

    $$-1=-1$$

11. Put the simplified groups together.

    $$x^2-x-6$$

12. Collect terms with the same power and simplify their coefficients.

    $$x^2-x-6$$

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

    $$x^2-x-6=0$$

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

    $$(2-3=-1,2\times \left(-3\right)=-6)$$

15. Rewrite the middle term using those two parts.

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

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

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

17. The common bracket can now be factored out.

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

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

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

19. Take this factor equal to zero.

    $$x-3=0$$

20. Subtract the constant from both sides.

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

21. Simplify both sides.

    $$x=3$$

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

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

23. Reduce the fraction to obtain this root.

    $$x=3$$

24. Take this factor equal to zero.

    $$x+2=0$$

25. Subtract the constant from both sides.

    $$x+2-2=0-2$$

26. Simplify both sides.

    $$x=-2$$

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

    $$x=-2\times \frac{1}{1}$$

28. Reduce the fraction to obtain this root.

    $$x=-2$$

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

    $$0=0$$

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

    $$0=0$$

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

    $$0=0$$

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

    $$0=0$$

Answer

The final answer is shown below.