Problem

9. Determine the equation of the quadratic function that passes through $(2,5)$ if the roots of the corresponding quadratic equation are $1+\sqrt{5}$ and $1-\sqrt{5}$.

Answer

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Answer

Final Answer: \(\boxed{f(x) = -1.25(x - (1+\sqrt{5}))(x - (1-\sqrt{5}))}\)

Steps

Step 1 ：Given roots of the quadratic equation are \(1 + \sqrt{5}\) and \(1 - \sqrt{5}\)

Step 2 ：Write the quadratic function in the form \(f(x) = a(x - r_1)(x - r_2)\)

Step 3 ：Plug in the point \((2, 5)\) to find the value of \(a\)

Step 4 ：Solve for \(a\) to get \(a = -1.25\)

Step 5 ：Final Answer: \(\boxed{f(x) = -1.25(x - (1+\sqrt{5}))(x - (1-\sqrt{5}))}\)

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