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Answer
The equation \(31x = 12x^2 + 20\) can be factored as \((4x - 5)(3x - 4) = 0\).
Solution
1. Start with the equation:
\[
31x = 12x^2 + 20
\]
2. Rearrange the equation by subtracting \(31x\) from both sides:
\[
12x^2 - 31x + 20 = 0
\]
3. Multiply the coefficient of \(x^2\) and the constant term:
\[
12 \times 20 = 240
\]
We need two numbers that multiply to \(240\) and add to \(-31\). The numbers \(-15\) and \(-16\) work because:
\[
(-15) \times (-16) = 240 \quad \text{and} \quad (-15) + (-16) = -31
\]
4. Rewrite the middle term using \(-15x\) and \(-16x\):
\[
12x^2 - 15x - 16x + 20 = 0
\]
5. Factor by grouping:
\[
(12x^2 - 15x) + (-16x + 20) = 0
\]
Factor out the common factors in each group:
\[
3x(4x - 5) - 4(4x - 5) = 0
\]
6. Factor out the common binomial factor \((4x - 5)\):
\[
(4x - 5)(3x - 4) = 0
\]
The factored form is:
\[
(4x - 5)(3x - 4) = 0
\]
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Simplify this solution Mind Expander
To factor the equation \( 12x^2 - 31x + 20 = 0 \), we start by rearranging it into standard quadratic form. We can use the quadratic formula to find the roots: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 12, b = -31, c = 20 \). Plugging in the values gives us the roots, which can then be used to express the quadratic in factored form. For common mistakes, remember to double-check your calculations when applying the quadratic formula. It's easy to make a small error in signs or arithmetic, which can lead you to the wrong conclusion. Always verify your roots by plugging them back into the original equation!
