To solve the equation [tex]\( x^2 = 5 - x \)[/tex] using the quadratic formula, let's follow the steps below in a detailed manner:
1. Rewrite the equation in standard form:
[tex]\[
x^2 + x - 5 = 0
\][/tex]
In this form, we can identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
a = 1, \quad b = 1, \quad c = -5
\][/tex]
2. Apply the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[
x = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-5)}}{2(1)}
\][/tex]
4. Simplify inside the square root:
[tex]\[
b^2 - 4ac = 1^2 - 4(1)(-5) = 1 + 20 = 21
\][/tex]
5. Plug this back into the formula:
[tex]\[
x = \frac{-1 \pm \sqrt{21}}{2}
\][/tex]
Thus, the solutions to the equation are:
[tex]\[
x = \frac{-1 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{21}}{2}
\][/tex]
Conclusion:
The values of [tex]\( x \)[/tex] that satisfy the equation [tex]\(x^2 = 5 - x\)[/tex] are:
[tex]\[
\boxed{\frac{-1 \pm \sqrt{21}}{2}}
\][/tex]
