To find the solutions to the quadratic equation [tex]\( 6 = x^2 - 10x \)[/tex], let's follow these steps:
1. Rewrite the Equation in Standard Form:
A quadratic equation in standard form is [tex]\( ax^2 + bx + c = 0 \)[/tex].
The given equation:
[tex]\[
6 = x^2 - 10x
\][/tex]
can be rewritten as:
[tex]\[
x^2 - 10x - 6 = 0
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -6 \)[/tex].
2. Identify the Coefficients:
From the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], identify the coefficients:
[tex]\[
a = 1,\quad b = -10,\quad c = -6
\][/tex]
3. Use the Quadratic Formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Plug in the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1}
\][/tex]
4. Simplify Inside the Square Root:
Calculate the discriminant [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
b^2 = (-10)^2 = 100
\][/tex]
[tex]\[
4ac = 4 \cdot 1 \cdot (-6) = -24
\][/tex]
[tex]\[
b^2 - 4ac = 100 - (-24) = 100 + 24 = 124
\][/tex]
5. Substitute Back into the Formula:
So the formula becomes:
[tex]\[
x = \frac{10 \pm \sqrt{124}}{2}
\][/tex]
6. Simplify the Radical:
Express [tex]\( \sqrt{124} \)[/tex] in simplest radical form:
[tex]\[
\sqrt{124} = \sqrt{4 \cdot 31} = 2\sqrt{31}
\][/tex]
Thus, the equation simplifies to:
[tex]\[
x = \frac{10 \pm 2\sqrt{31}}{2}
\][/tex]
7. Simplify the Fraction:
Divide both terms in the numerator by 2:
[tex]\[
x = 5 \pm \sqrt{31}
\][/tex]
Therefore, the solutions in simplest radical form are:
[tex]\[
x = 5 + \sqrt{31} \quad \text{and} \quad x = 5 - \sqrt{31}
\][/tex]
