Question

Solve the equation by factoring. $$30 x^{2}-80 x+50=7 x-4$$

Short answer

The solution to the equation are x = 1.2 and x = 1.5

Step-by-step solution

- Rearranging the Equation

First, we collect like terms. This means getting the terms with the variable x on one side and constant terms on the other side. We subtract \(7x - 4\) from both sides to maintain equality and get \(30x^{2} - 80x - 7x + 50 + 4 = 0\). This simplifies to \(30x^{2} - 87x + 54 = 0 \)

- Factoring the Equation

Next, we need to factorize the obtained equation. For this, we seek two numbers that multiply to give \(30 * 54 = 1620\) (product of the coefficient of \(x^2\) and the constant term) and add up to give -87 (coefficient of x). After factoring we find such numbers to be -36 and -45. That is, \(-36 * -45 = 1620\) and \(-36 - 45 = -87\). The equation can now be expressed as \(30x^{2} - 36x - 51x + 54 = 0\), which simplifies to \(6x(5x - 6) - 9(5x - 6) = 0\).

- Solve the Equation for x

Now, we'll solve for x. We will set each individual factor to zero and solve. From \(6x(5x - 6) - 9(5x - 6) = 0\), we get that \(5x - 6 = 0 \) or \(6x - 9 = 0\). Solving these gives us x = 1.2 and x = 1.5 as the solutions.