Question

what are the solutions of the equation \\((2x + 3)^2 + 8(2x + 3) + 11 = 0\\)? use \\(u\\) substitution and the quadratic formula to solve.

\\(x = \frac{-4 \pm \sqrt{5}}{2}\\)

\\(x = \frac{-7 \pm \sqrt{5}}{2}\\)

\\(x = -7\\) and \\(x = -2\\)

\\(x = -1\\) and \\(x = 4\\)

Explanation:

Apply u-substitution

Using the Quadratic Form Equations knowledge point

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Solve for u using the quadratic formula

Using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = 8\), and \(c = 11\):

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Back-substitute to solve for x

Substitute \(u = 2x + 3\) back into the equation:

$$
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Answer:

• (A) \(x = \frac{-4 \pm \sqrt{5}}{2}\)
• (B) \(x = \frac{-7 \pm \sqrt{5}}{2}\) (Correct answer)
• (C) \(x = -7 \text{ and } x = -2\)
• (D) \(x = -1 \text{ and } x = 4\)