Question

Solve the equation by factoring. $$5 x^{2}-3 x-26=0$$

Short answer

The solutions to the equation are x = -13/5 and x = 2.

Step-by-step solution

Identify the quadratic equation

The given equation is \(5x^2-3x-26=0\) in the form \(ax^2+bx+c=0\), where \(a=5\), \(b=-3\), and \(c=-26\).

Factor the quadratic equation

First, factors of \(ac\) or \(5*(-26)=-130\) are found which add or subtract to give \(b=-3\). These factors are -10 and 13. That is, \(-10+13=-3\). Now, equation is rewritten by splitting the middle term: \(5x^2 -10x + 13x -26 =0\). Taking out common terms, we get: \(5x(x-2) + 13(x-2)=0\). So, the factored form of the equation is \((5x+13)(x-2)=0\).

Find the roots

A product of factors equals zero only if at least one of the factors is zero. Therefore, set each factor equal to zero and solve for \(x\): 5x+13=0 implies x=-13/5, and x-2=0 implies x=2.