Question

Use the quadratic formula to solve the equation. $$y^{2}+11 y+10=0$$

Short answer

The solutions to the equation \(y^{2}+11y+10=0\) are \(y = -1\) and \(y = -10\).

Step-by-step solution

Identification of coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. In this case, \(a=1\), \(b=11\), and \(c=10\).

Application of Quadratic Formula

Apply the quadratic formula \(y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Substituting the identified values of \(a\), \(b\), and \(c\) into the formulas gives us: \(y = \frac{-11 \pm \sqrt{11^{2} - 4*1*10}}{2*1}\). This simplifies to \(y = \frac{-11 \pm \sqrt{121 - 40}}{2}\), which leads to \(y = \frac{-11 \pm \sqrt{81}}{2}\). Thus, the formula simplifies to \(y = \frac{-11 \pm 9}{2}\).

Calculate the roots

Finally, we calculate the two possible values of \(y\). For \(y = \frac{-11 + 9}{2}\), the answer is -1. Similarly, for \(y = \frac{-11 - 9}{2}\), the answer is -10. Therefore, the solutions to the quadratic equation are \(y = -1\) and \(y = -10\).