Question

Solve x^2 + 4x = 27

Answer 1

To solve the equation \(x^2 + 4x = 27\), we first set it to zero by subtracting 27 from both sides:

\[
x^2 + 4x - 27 = 0
\]

Next, we can use the quadratic formula to find the values of \(x\). The quadratic formula is given by:

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

In our equation \(x^2 + 4x - 27 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 4\)
- \(c = -27\)

Now, we can substitute these values into the quadratic formula:

1. Calculate the discriminant:

\[
b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-27) = 16 + 108 = 124
\]

2. Use the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{124}}{2 \cdot 1}
\]

3. Simplify \(\sqrt{124}\):

\[
\sqrt{124} = \sqrt{4 \cdot 31} = 2\sqrt{31}
\]

4. Substitute back into the formula:

\[
x = \frac{-4 \pm 2\sqrt{31}}{2}
\]

5. Simplify:

\[
x = -2 \pm \sqrt{31}
\]

Thus, the solutions for the equation \(x^2 + 4x = 27\) are:

\[
x = -2 + \sqrt{31} \quad \text{and} \quad x = -2 - \sqrt{31}
\]