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Factor this \( 31 x=12 x^{2}+20 \)

Ask by Lang Burns. in the United States
Mar 19,2025

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Answer

The equation \( 31x = 12x^2 + 20 \) can be factored as \( (4x - 5)(3x - 4) = 0 \).

Solution

We start with the equation \[ 31x = 12x^2 + 20. \] First, bring all terms to one side to obtain a standard quadratic equation: \[ 12x^2 - 31x + 20 = 0. \] Next, we look for two numbers whose product is \[ 12 \times 20 = 240 \] and whose sum is \(-31\). Notice that \(-15\) and \(-16\) work because \[ (-15) \times (-16) = 240 \quad \text{and} \quad (-15) + (-16) = -31. \] Now, rewrite the middle term using \(-15x\) and \(-16x\): \[ 12x^2 - 15x - 16x + 20 = 0. \] Group the terms: \[ (12x^2 - 15x) + (-16x + 20) = 0. \] Factor out the common factors from each group: \[ 3x(4x - 5) - 4(4x - 5) = 0. \] Now, factor by grouping: \[ (4x - 5)(3x - 4) = 0. \] Thus, the factored form of the equation is: \[ (4x - 5)(3x - 4) = 0. \]

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The Deep Dive

To factor the equation \( 31x = 12x^2 + 20 \), first, move all terms to one side to set the equation to zero: \( 12x^2 - 31x + 20 = 0 \). Now, look for two numbers that multiply to \( 12 \times 20 = 240 \) and add up to \( -31 \). Those numbers are \( -16 \) and \( -15 \). Rewrite the equation using these numbers: \( 12x^2 - 16x - 15x + 20 = 0 \). Now group the terms: \( 4x(3x - 4) - 5(3x - 4) = 0 \). Factor by grouping: \( (3x - 4)(4x - 5) = 0 \). Thus, the factored form is \( (3x - 4)(4x - 5) = 0 \). You can solve for \( x \) by setting each factor to zero: \( 3x - 4 = 0 \) results in \( x = \frac{4}{3} \) and \( 4x - 5 = 0 \) results in \( x = \frac{5}{4} \). Happy factoring!

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