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Factor the following expression completely: 6w^2-7w+2=square

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Factor the following expression completely:
6w^2-7w+2=square

Factor the following expression completely: 6w^2-7w+2=square

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Normaveterano · Tutor durante 12 años
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To factor the quadratic expression \(6w^2 - 7w + 2\) completely, we need to find two binomials whose product gives us the original quadratic expression.<br /><br />The general form of a factored quadratic expression is:<br />\[ (aw + b)(cw + d) \]<br /><br />We need to find \(a\), \(b\), \(c\), and \(d\) such that:<br />\[ ac = 6 \]<br />\[ bd = 2 \]<br />\[ ad + bc = -7 \]<br /><br />Let's try to find the correct pair of factors.<br /><br />First, list the pairs of factors of 6:<br />\[ (1, 6) \]<br />\[ (2, 3) \]<br /><br />Next, list the pairs of factors of 2:<br />\[ (1, 2) \]<br /><br />Now, we need to find a combination that satisfies all the conditions. Let's try:<br />\[ (3w - 2)(2w - 1) \]<br /><br />Let's verify:<br />\[ (3w - 2)(2w - 1) = 3w \cdot 2w + 3w \cdot (-1) + (-2) \cdot 2w + (-2) \cdot (-1) \]<br />\[ = 6w^2 - 3w - 4w + 2 \]<br />\[ = 6w^2 - 7w + 2 \]<br /><br />This matches the original expression. Therefore, the factored form is:<br />\[ 6w^2 - 7w + 2 = (3w - 2)(2w - 1) \]<br /><br />So, the completely factored form of the expression is:<br />\[ \boxed{(3w - 2)(2w - 1)} \]
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