Factor this $$ 31 x=12 x^{2}+20 $$

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UpStudy AI · Accepted Answer

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

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

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