Mathematics 10 — Exercise 2.1 · Q2 (ii)

Full question

Solve the following quadratic equation by factorization method: x²+3 × x-28 = 0.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$x^2+3\times x-28-0$$

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

   $$x^2+3\times x-28$$

3. Write EVERY distributed product before collecting terms.

   $$-28+x^2+3\times x-0$$

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

   $$-28=-28$$

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

   $$x^2=x^2$$

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

   $$3\times x=3\times x$$

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.

   $$-28+0=-28$$

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.

    $$3=3$$

11. Put the simplified groups together.

    $$x^2+3\times x-28$$

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

    $$x^2+3\times x-28$$

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

    $$x^2+3\times x-28=0$$

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

    $$(7-4=3,7\times \left(-4\right)=-28)$$

15. Rewrite the middle term using those two parts.

    $$x^2+7\times x-4\times x-28=0$$

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

    $$x\times \left(x+7\right)-4\times \left(x+7\right)=0$$

17. The common bracket can now be factored out.

    $$\left(x-4\right)\times \left(x+7\right)=0$$

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

    $$(x-4=0,x+7=0)$$

19. Take this factor equal to zero.

    $$x-4=0$$

20. Subtract the constant from both sides.

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

21. Simplify both sides.

    $$x=4$$

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

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

23. Reduce the fraction to obtain this root.

    $$x=4$$

24. Take this factor equal to zero.

    $$x+7=0$$

25. Subtract the constant from both sides.

    $$x+7-7=0-7$$

26. Simplify both sides.

    $$x=-7$$

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

    $$x=-7\times \frac{1}{1}$$

28. Reduce the fraction to obtain this root.

    $$x=-7$$

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.