Answer :

To solve the equation [tex]\(x(x + 4) = 221\)[/tex], follow these steps:

1. Expand the Equation:
Start by expanding the left side of the equation:
[tex]\[
x(x + 4) = x^2 + 4x
\][/tex]
So the equation becomes:
[tex]\[
x^2 + 4x = 221
\][/tex]

2. Make It a Standard Form Quadratic Equation:
Bring all terms to one side to set the equation to zero:
[tex]\[
x^2 + 4x - 221 = 0
\][/tex]

3. Solve the Quadratic Equation:
Now, solve the quadratic equation [tex]\(x^2 + 4x - 221 = 0\)[/tex]. This can be done using the quadratic formula, where [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -221\)[/tex]. The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]

4. Calculate the Discriminant:
First, compute the discriminant ([tex]\(b^2 - 4ac\)[/tex]):
[tex]\[
b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-221) = 16 + 884 = 900
\][/tex]

5. Find the Roots:
Now substitute the values into the quadratic formula:
[tex]\[
x = \frac{-4 \pm \sqrt{900}}{2}
\][/tex]
[tex]\[
x = \frac{-4 \pm 30}{2}
\][/tex]
This gives two solutions:
[tex]\[
x = \frac{-4 + 30}{2} = \frac{26}{2} = 13
\][/tex]
and
[tex]\[
x = \frac{-4 - 30}{2} = \frac{-34}{2} = -17
\][/tex]

Therefore, the solutions for the equation [tex]\(x(x + 4) = 221\)[/tex] are [tex]\(x = 13\)[/tex] and [tex]\(x = -17\)[/tex].

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