Questions: Use the quadratic formula to find the real solutions, if any, of the equation.
[ x^2+2 x-13=0 ]
Transcript text: point(s) possible
Use the quadratic formula to find the real solutions, if any, of the equation.
\[
x^{2}+2 x-13=0
\]
Solution
Solution Steps
To solve the quadratic equation \(x^2 + 2x - 13 = 0\) using the quadratic formula, we need to identify the coefficients \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\). Then, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the real solutions.
Step 1: Identify the coefficients
Given the quadratic equation \(x^2 + 2x - 13 = 0\), we identify the coefficients:
\[
a = 1, \quad b = 2, \quad c = -13
\]
Step 2: Calculate the discriminant
The discriminant \(\Delta\) of the quadratic equation is given by:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \(a\), \(b\), and \(c\):
\[
\Delta = 2^2 - 4 \cdot 1 \cdot (-13) = 4 + 52 = 56
\]
Step 3: Apply the quadratic formula
The quadratic formula to find the roots of the equation \(ax^2 + bx + c = 0\) is:
\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\]
Substituting the values of \(a\), \(b\), and \(\Delta\):
\[
x = \frac{-2 \pm \sqrt{56}}{2 \cdot 1} = \frac{-2 \pm 7.4833}{2}
\]
Step 4: Calculate the roots
Calculating the two possible values for \(x\):
\[
x_1 = \frac{-2 + 7.4833}{2} = \frac{5.4833}{2} = 2.7417
\]
\[
x_2 = \frac{-2 - 7.4833}{2} = \frac{-9.4833}{2} = -4.7417
\]