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(a) Use the quadratic formula to solve \( x^{2}-7 x+3=0 \). Separate multiple answers with a comma, if necessary. Write your answer in exact, simplified form. The solution set is \( \{\square\} \). S

Ask by Pollard Hilton. in the United States
Mar 16,2025

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The solution set is \( \left\{\frac{7+\sqrt{37}}{2}, \frac{7-\sqrt{37}}{2}\right\} \).

Solución

1. Start with the quadratic equation: \[ x^2 - 7x + 3 = 0. \] 2. Identify the coefficients: \( a = 1 \), \( b = -7 \), and \( c = 3 \). 3. Write the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] 4. Substitute the values of \( a \), \( b \), and \( c \) into the formula: \[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(3)}}{2(1)}. \] 5. Simplify the expression: \[ x = \frac{7 \pm \sqrt{49 - 12}}{2}, \] \[ x = \frac{7 \pm \sqrt{37}}{2}. \] 6. The solution set is: \[ \left\{\frac{7+\sqrt{37}}{2}, \frac{7-\sqrt{37}}{2}\right\}. \]

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The Deep Dive

To solve the quadratic equation \( x^{2}-7x+3=0 \) using the quadratic formula, we start with the formula itself: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = -7 \), and \( c = 3 \). First, we calculate the discriminant: \[ b^2 - 4ac = (-7)^2 - 4(1)(3) = 49 - 12 = 37 \] Now, plugging the values into the formula gives: \[ x = \frac{-(-7) \pm \sqrt{37}}{2(1)} = \frac{7 \pm \sqrt{37}}{2} \] Thus, the solution set is: \[ \left\{ \frac{7 + \sqrt{37}}{2}, \frac{7 - \sqrt{37}}{2} \right\} \]

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