Question

Multiply each side of the equation by an appropriate power of ten to obtain integer coefficients. Then solve by factoring. $$0.8 x^{2}+3.2 x+2.40=0$$

Short answer

After converting decimal coefficients to integers and factoring, the solutions for \(x\) in the equation are \(x = -1.5\) and \(x = -2\).

Step-by-step solution

Identify the smallest decimal place in the equation

In this equation, we see that the smallest decimal place is 2. Hence, we need to multiply the entire equation by \(10^{2}\) to get integer coefficients.

Multiply the equation by an appropriate power of ten

Multiply each term by \(10^{2}\) (or 100). We get: \(80x^{2} + 320x + 240 = 0\).

Factor the equation

The equation can be factored as \(40(2x^{2} + 8x + 6) = 0\). This equation can be further factored to \(40(2x + 3)(x + 2) = 0\).

Solve for x

Setting each factor equal to zero and solving for x, we obtain \(2x + 3 = 0\) that gives \(x = -1.5\), and \(x + 2 = 0\) that gives \(x = -2\).