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Question Factor completely. 5x^2-36x+7 square

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Factor completely.
5x^2-36x+7
square

Question Factor completely. 5x^2-36x+7 square

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AfonsoVeterano · Tutor por 11 anos

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To factor the quadratic expression \(5x^2 - 36x + 7\) completely, we need to find two binomials whose product gives us the original quadratic expression.<br /><br />The general form of factoring a quadratic expression \(ax^2 + bx + c\) is:<br />\[ (mx + n)(px + q) \]<br /><br />For our specific quadratic expression \(5x^2 - 36x + 7\), we need to find two numbers that multiply to \(a \cdot c = 5 \cdot 7 = 35\) and add up to \(b = -36\).<br /><br />Let's denote these two numbers as \(m\) and \(n\). We need:<br />\[ m \cdot n = 35 \]<br />\[ m + n = -36 \]<br /><br />By examining the pairs of factors of 35, we find:<br />- \(1 \cdot 35 = 35\) and \(1 + 35 = 36\)<br />- \(-1 \cdot -35 = 35\) and \(-1 + (-35) = -36\)<br /><br />Thus, the correct pair is \(-1\) and \(-35\).<br /><br />Now, we rewrite the middle term \(-36x\) using these two numbers:<br />\[ 5x^2 - 36x + 7 = 5x^2 - 1x - 35x + 7 \]<br /><br />Next, we group the terms:<br />\[ = (5x^2 - 1x) + (-35x + 7) \]<br /><br />We factor out the greatest common factor (GCF) from each group:<br />\[ = x(5x - 1) - 7(5x - 1) \]<br /><br />Now, we factor out the common binomial factor \((5x - 1)\):<br />\[ = (5x - 1)(x - 7) \]<br /><br />Therefore, the completely factored form of \(5x^2 - 36x + 7\) is:<br />\[ \boxed{(5x - 1)(x - 7)} \]
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