Question: 7 — Completing the Square
LEARNING OBJECTIVE: Determine an equivalent quadratic equation by
completing the square.
Which of the following is equivalent to 0 = 2x^2 - 16x - 18 when completing the
square?
a.) (x - 4)^2 = 34
b.) (x - 4)^2 = 25
c.) (x - 8)^2 = 34
d.) (x - 8)^2 = 25
1. **State the problem:** We want to find an equivalent quadratic equation to $$0 = 2x^2 - 16x - 18$$ by completing the square.
2. **Start by isolating the quadratic and linear terms:**
$$2x^2 - 16x = 18$$
3. **Factor out the coefficient of $$x^2$$ from the left side:**
$$2(x^2 - 8x) = 18$$
4. **Complete the square inside the parentheses:**
Take half of the coefficient of $$x$$, which is $$-8$$, half is $$-4$$, then square it:
$$(-4)^2 = 16$$
5. **Add and subtract 16 inside the parentheses to complete the square:**
$$2(x^2 - 8x + 16 - 16) = 18$$
6. **Rewrite the expression grouping the perfect square trinomial:**
$$2((x - 4)^2 - 16) = 18$$
7. **Distribute the 2:**
$$2(x - 4)^2 - 32 = 18$$
8. **Add 32 to both sides:**
$$2(x - 4)^2 = 18 + 32$$
$$2(x - 4)^2 = 50$$
9. **Divide both sides by 2:**
$$\cancel{2}(x - 4)^2 = \cancel{2} \times 25$$
$$ (x - 4)^2 = 25$$
**Final answer:** $$ (x - 4)^2 = 25 $$ which corresponds to option b.)