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Question Factor completely. 5x^2-12x+7 Answer Allempi

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Factor completely.
5x^2-12x+7
Answer Allempi

Question Factor completely. 5x^2-12x+7 Answer Allempi

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LucasProfissional · Tutor por 6 anos

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To factor the quadratic expression \(5x^2 - 12x + 7\) completely, we need to find two binomials whose product gives us the original quadratic expression.<br /><br />First, we identify the coefficients:<br />- \(a = 5\)<br />- \(b = -12\)<br />- \(c = 7\)<br /><br />Next, we use the quadratic formula to find the roots of the quadratic equation \(5x^2 - 12x + 7 = 0\):<br /><br />\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br /><br />Substituting the values of \(a\), \(b\), and \(c\):<br /><br />\[ x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 5 \cdot 7}}{2 \cdot 5} \]<br />\[ x = \frac{12 \pm \sqrt{144 - 140}}{10} \]<br />\[ x = \frac{12 \pm \sqrt{4}}{10} \]<br />\[ x = \frac{12 \pm 2}{10} \]<br /><br />This gives us two solutions:<br /><br />\[ x = \frac{12 + 2}{10} = \frac{14}{10} = \frac{7}{5} \]<br />\[ x = \frac{12 - 2}{10} = \frac{10}{10} = 1 \]<br /><br />So, the roots are \(x = \frac{7}{5}\) and \(x = 1\).<br /><br />Now, we can write the quadratic expression as a product of two binomials using these roots:<br /><br />\[ 5x^2 - 12x + 7 = 5(x - 1)\left(x - \frac{7}{5}\right) \]<br /><br />To simplify, we can factor out the 5 from the second binomial:<br /><br />\[ 5x^2 - 12x + 7 = 5(x - 1)\left(\frac{5x - 7}{5}\right) \]<br />\[ 5x^2 - 12x + 7 = (x - 1)(5x - 7) \]<br /><br />Thus, the completely factored form of \(5x^2 - 12x + 7\) is:<br /><br />\[ \boxed{(x - 1)(5x - 7)} \]
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