Question

Solve the equation. Tell which solution method you used. \(y^{2}+7 y+12=0\)

Short answer

The solutions to the quadratic equation \(y^{2}+7y+12=0\) are \(y=-3\) and \(y=-4\). The quadratic formula was used to find these solutions.

Step-by-step solution

Identification of the coefficients

The first step is to identify the coefficients a, b, and c in the equation \(y^{2}+7y+12=0\). Here, \(a=1\), \(b=7\), and \(c=12\).

Substitute the coefficients into the quadratic formula

Substitute the values of a, b, and c respectively into the quadratic formula. This gives \(\frac{{-7 \pm \sqrt{{(7)^{2}-4(1)(12)}}}}{2(1)}\). This simplifies to \(\frac{{-7 \pm \sqrt{{49-48}}}}{2}\) and further simplifies to \(\frac{{-7 \pm \sqrt{1}}}{2}\).

Determine the roots of the equation

Evaluate each part of the equation separately. The equation \(\frac{{-7+1}}{2} = -3\) and \(\frac{{-7-1}}{2} = -4\)\ are the roots (or solutions) of the equation after simplifying.