Question

diagnostic assessment

1. for the equation (x^{2}+8x = 17), use any algebraic method to determine the exact solution.
2. explain how converting a quadratic equation to vertex form helps identify the vertex coordinates and other key features of the associated quadratic function. provide specific examples to illustrate your explanation.

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Explanation:

Step1: Complete the square

Given \(x^{2}+8x = 17\).
For the quadratic \(ax^{2}+bx + c\) (here \(a = 1\), \(b=8\)), the term to add to complete the square is \((\frac{b}{2})^{2}\).
Since \(b = 8\), \((\frac{8}{2})^{2}=16\).
Add \(16\) to both sides of the equation:
\(x^{2}+8x + 16=17 + 16\).
Using the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\) (here \(a=x\), \(b = 4\)), we get \((x + 4)^{2}=33\).

Step2: Solve for \(x\)

Take the square root of both sides:
\(x+4=\pm\sqrt{33}\).
Subtract \(4\) from both sides:
\(x=-4\pm\sqrt{33}\).

Answer:

\(x=-4+\sqrt{33}\) and \(x=-4 - \sqrt{33}\)