To solve the equation \(6x^2 + 7x = 12\) by factoring, we need to rearrange the equation to make it equal to zero:
\(6x^2 + 7x - 12 = 0\)
To factor the quadratic expression \(6x^2 + 7x - 12\), we look for two numbers that multiply to the product of the leading coefficient (6) and the constant term (-12), which is -72, and add up to the coefficient of the x-term (7). The two numbers that meet these criteria are 9 and -8.
Therefore, we rewrite the middle term (7x) using these two numbers:
\(6x^2 + 9x - 8x - 12 = 0\)
Now, we factor by grouping:
\(3x(2x + 3) - 4(2x + 3) = 0\)
Now, we factor out the common binomial factor:
\((3x - 4)(2x + 3) = 0\)
Next, we set each factor to zero and solve for x:
\(3x - 4 = 0\) or \(2x + 3 = 0\)
\(3x = 4\) or \(2x = -3\)
\(x = \frac{4}{3}\) or \(x = -\frac{3}{2}\)
Therefore, the solutions to the equation \(6x^2 + 7x = 12\) by factoring are \(x = \frac{4}{3}\) and \(x = -\frac{3}{2}\).
