Pregunta
(b) Write \( x^{2}-7 x+3 \) as a product of linear factors. Write your answer in exact, simplified form. \( x^{2}-7 x+3=\square \)
Ask by Watkins Fleming. in the United States
Mar 16,2025
Solución de inteligencia artificial de Upstudy
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\( x^{2}-7x+3 = \left(x - \frac{7 + \sqrt{37}}{2}\right)\left(x - \frac{7 - \sqrt{37}}{2}\right) \)
Solución
We start with the quadratic expression
\[
x^{2}-7x+3.
\]
1. Identify the coefficients for the quadratic formula:
\[
a=1,\quad b=-7,\quad c=3.
\]
2. Compute the discriminant:
\[
\Delta = b^2 - 4ac = (-7)^2 - 4(1)(3) = 49-12 = 37.
\]
3. Use the quadratic formula to find the roots:
\[
x = \frac{-b\pm\sqrt{\Delta}}{2a} = \frac{7\pm\sqrt{37}}{2}.
\]
4. Express the quadratic as a product of its linear factors using the roots:
\[
x^{2}-7x+3 = \left(x-\frac{7+\sqrt{37}}{2}\right)\left(x-\frac{7-\sqrt{37}}{2}\right).
\]
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Simplifique esta solución The Deep Dive
To factor \( x^{2}-7x+3 \) into linear factors, we can use the quadratic formula, which is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -7 \), and \( c = 3 \). Plugging these values into the formula, we have: \[ x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} \] \[ x = \frac{7 \pm \sqrt{49 - 12}}{2} \] \[ x = \frac{7 \pm \sqrt{37}}{2} \] Thus, the roots of the quadratic equation are \( \frac{7 + \sqrt{37}}{2} \) and \( \frac{7 - \sqrt{37}}{2} \). The factorization can be written as: \[ x^{2}-7x+3 = \left(x - \frac{7 + \sqrt{37}}{2}\right)\left(x - \frac{7 - \sqrt{37}}{2}\right) \]
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