Questions: Solve the equation using the quadratic formula. x^2+12x+35=0

Solution Steps

To solve the quadratic equation \(x^2 + 12x + 35 = 0\) using the quadratic formula, we identify the coefficients \(a = 1\), \(b = 12\), and \(c = 35\). The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

We will substitute the values of \(a\), \(b\), and \(c\) into this formula to find the roots of the equation.

The given quadratic equation is

\[ x^2 + 12x + 35 = 0 \]

From this equation, we identify the coefficients as follows:

\[ a = 1, \quad b = 12, \quad c = 35 \]

We calculate the discriminant \(D\) using the formula

\[ D = b^2 - 4ac \]

Substituting the values of \(a\), \(b\), and \(c\):

\[ D = 12^2 - 4 \cdot 1 \cdot 35 = 144 - 140 = 4 \]

Using the quadratic formula

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

we substitute \(b = 12\), \(D = 4\), and \(a = 1\):

\[ x = \frac{-12 \pm \sqrt{4}}{2 \cdot 1} = \frac{-12 \pm 2}{2} \]

This gives us two solutions:

\[ x_1 = \frac{-12 + 2}{2} = \frac{-10}{2} = -5 \]

\[ x_2 = \frac{-12 - 2}{2} = \frac{-14}{2} = -7 \]

Final Answer