Question

Solve the equation by factoring. $$18 x^{2}-30 x-100=67 x+30$$

Short answer

The solutions to the equation \(18x^{2} - 30x - 100 = 67x + 30\) are \(x = -\frac{13}{6}\) and \(x = \frac{10}{3}\)

Step-by-step solution

Rewrite the equation

To rewrite the equation \(18x^{2} - 30x - 100 = 67x + 30\) in the standard form, move all the terms to the left side of the equation: \(18x^{2} - 30x - 67x - 100 - 30 = 0\). Combine like terms to get: \(18x^{2} - 97x - 130 = 0\)

Factoring the equation

Now that the equation is in standard form, we can proceed to factoring. We can also factor out a common factor among all three terms on the left side, which is 1 in this case to keep the equation simplified. As there's not a single simple factor that can be taken out of all three terms, we skip to finding two numbers that add up to -97 (the coefficient of x) and multiply to -2340 (the product of 18 and -130). Those numbers are -60 and 39 because -60 + 39 = -21 and -60 * 39 = -2340. So, rewrite -97x as -60x + 39x to get: \(18x^{2} - 60x + 39x - 130 = 0\) . Now factor by grouping to get: \(6x(3x - 10) +13(3x - 10) = 0\)

Solve for x

The final factored form of the equation is \((6x + 13)(3x - 10) = 0\). To find the roots, set each factor equal to zero and solve for x. This gives us two solutions: \(x = -\frac{13}{6}\) and \(x = \frac{10}{3}\)