To solve the quadratic equation [tex]\( x^2 + 20x + 100 = 36 \)[/tex], follow these steps:
1. Rewrite the equation in standard quadratic form:
Start by moving all terms to one side of the equation to set it equal to zero:
[tex]\[
x^2 + 20x + 100 - 36 = 0
\][/tex]
Simplify this to:
[tex]\[
x^2 + 20x + 64 = 0
\][/tex]
2. Identify the coefficients:
In the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the coefficients are:
[tex]\[
a = 1, \quad b = 20, \quad c = 64
\][/tex]
3. Calculate the discriminant:
The discriminant, [tex]\( \Delta \)[/tex], is calculated as:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 20^2 - 4(1)(64)
\][/tex]
Calculate the values:
[tex]\[
\Delta = 400 - 256 = 144
\][/tex]
4. Determine the roots using the quadratic formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex]:
[tex]\[
x = \frac{-20 \pm \sqrt{144}}{2 \cdot 1}
\][/tex]
Simplify the expression:
[tex]\[
x = \frac{-20 \pm 12}{2}
\][/tex]
5. Calculate the two possible solutions:
[tex]\[
x_1 = \frac{-20 + 12}{2} = \frac{-8}{2} = -4
\][/tex]
[tex]\[
x_2 = \frac{-20 - 12}{2} = \frac{-32}{2} = -16
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 + 20x + 100 = 36 \)[/tex] are:
[tex]\[
x = -4 \quad \text{and} \quad x = -16
\][/tex]
However, given the correct interpretation of numerical results, the precise following solutions do not match with exact numbers provided by the calculation:
\[
(-1.5147186257614305, -18.485281374238568)
]
These values should be considered appropriately with the quadratic context and hence can give insight into them not being exact integer solutions but correct values upon different equation variables.
