To solve the equation \(x^2 - 4x - 9 = 29\), follow these steps:
1. Rearrange the equation:
[tex]\[
x^2 - 4x - 9 = 29
\][/tex]
Move 29 to the left side to set the equation to equal zero:
[tex]\[
x^2 - 4x - 9 - 29 = 0
\][/tex]
Simplify the equation:
[tex]\[
x^2 - 4x - 38 = 0
\][/tex]
2. Recognize the form:
This is a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where:
\(a = 1\),
\(b = -4\),
\(c = -38\).
3. Use the quadratic formula to solve for \(x\):
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Substitute the values of \(a\), \(b\), and \(c\):
[tex]\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-38)}}{2(1)}
\][/tex]
Simplify the terms:
[tex]\[
x = \frac{4 \pm \sqrt{16 + 152}}{2}
\][/tex]
Combine the terms inside the square root:
[tex]\[
x = \frac{4 \pm \sqrt{168}}{2}
\][/tex]
5. Simplify the square root and the fraction:
Notice that:
[tex]\[
\sqrt{168} = \sqrt{4 \cdot 42} = 2\sqrt{42}
\][/tex]
So substitute back:
[tex]\[
x = \frac{4 \pm 2\sqrt{42}}{2}
\][/tex]
Factor out the 2 in the numerator:
[tex]\[
x = \frac{2(2 \pm \sqrt{42})}{2}
\][/tex]
Cancel out the 2:
[tex]\[
x = 2 \pm \sqrt{42}
\][/tex]
6. Interpret the final solutions:
[tex]\[
x = 2 + \sqrt{42}
\][/tex]
and
[tex]\[
x = 2 - \sqrt{42}
\][/tex]
So the solution for the equation \(x^2 - 4x - 9 = 29\) is:
[tex]\[
x = 2 \pm \sqrt{42}
\][/tex]
