QuestionAugust 25, 2025

7. Rewrite by completing the square. (a) x^2+x+1 (b) 2x^2-12x+11

7. Rewrite by completing the square. (a) x^2+x+1 (b) 2x^2-12x+11
7. Rewrite by completing the square. (a) x^2+x+1 (b) 2x^2-12x+11

Solution
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Answer

(a) (x + \frac{1}{2})^2 + \frac{3}{4} ### (b) 2(x - 3)^2 - 7 Explanation 1. Complete the square for x^2 + x + 1 Take half of the coefficient of x, square it, and add/subtract inside the expression: (x^2 + x + \frac{1}{4}) - \frac{1}{4} + 1. 2. Simplify the expression The expression becomes (x + \frac{1}{2})^2 + \frac{3}{4}. 3. Complete the square for 2x^2 - 12x + 11 Factor out the leading coefficient from the quadratic and linear terms: 2(x^2 - 6x) + 11. 4. Complete the square inside the parentheses Take half of the coefficient of x, square it, and add/subtract inside the expression: 2((x^2 - 6x + 9) - 9) + 11. 5. Simplify the expression The expression becomes 2(x - 3)^2 - 18 + 11 = 2(x - 3)^2 - 7.

Explanation

1. Complete the square for $x^2 + x + 1$<br /> Take half of the coefficient of $x$, square it, and add/subtract inside the expression: $(x^2 + x + \frac{1}{4}) - \frac{1}{4} + 1$.<br /><br />2. Simplify the expression<br /> The expression becomes $(x + \frac{1}{2})^2 + \frac{3}{4}$.<br /><br />3. Complete the square for $2x^2 - 12x + 11$<br /> Factor out the leading coefficient from the quadratic and linear terms: $2(x^2 - 6x) + 11$.<br /><br />4. Complete the square inside the parentheses<br /> Take half of the coefficient of $x$, square it, and add/subtract inside the expression: $2((x^2 - 6x + 9) - 9) + 11$.<br /><br />5. Simplify the expression<br /> The expression becomes $2(x - 3)^2 - 18 + 11 = 2(x - 3)^2 - 7$.
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