To solve the quadratic equation [tex]\( n^2 = 9n - 20 \)[/tex] using the quadratic formula, we first need to rewrite it in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Step 1: Rewrite the equation
[tex]\[
n^2 - 9n + 20 = 0
\][/tex]
From this equation, we identify the coefficients:
[tex]\[
a = 1, \quad b = -9, \quad c = 20
\][/tex]
Step 2: Use the quadratic formula
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Step 3: Calculate the discriminant
The discriminant ([tex]\(\Delta\)[/tex]) is part of the quadratic formula given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-9)^2 - 4 \cdot 1 \cdot 20
\][/tex]
[tex]\[
\Delta = 81 - 80
\][/tex]
[tex]\[
\Delta = 1
\][/tex]
Step 4: Compute the square root of the discriminant
[tex]\[
\sqrt{\Delta} = \sqrt{1} = 1.0
\][/tex]
Step 5: Calculate the two potential solutions using the quadratic formula
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} \quad \text{and} \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a}
\][/tex]
Substituting the known values:
[tex]\[
x_1 = \frac{-(-9) + 1.0}{2 \cdot 1}
\][/tex]
[tex]\[
x_1 = \frac{9 + 1.0}{2}
\][/tex]
[tex]\[
x_1 = \frac{10}{2}
\][/tex]
[tex]\[
x_1 = 5.0
\][/tex]
[tex]\[
x_2 = \frac{-(-9) - 1.0}{2 \cdot 1}
\][/tex]
[tex]\[
x_2 = \frac{9 - 1.0}{2}
\][/tex]
[tex]\[
x_2 = \frac{8}{2}
\][/tex]
[tex]\[
x_2 = 4.0
\][/tex]
Therefore, the solutions to the equation [tex]\( n^2 = 9n - 20 \)[/tex] are:
[tex]\[
n = 5.0 \quad \text{and} \quad n = 4.0
\][/tex]
So, the step-by-step solution yields the following results:
[tex]\[
\text{Discriminant: } 1
\][/tex]
[tex]\[
\text{Square Root of Discriminant: } 1.0
\][/tex]
[tex]\[
\text{Solutions: } n = 5.0 \text{ and } n = 4.0
\][/tex]
