Let's solve the equation step-by-step:
[tex]\[
-x(x-2) + x + (2x + 8) + 10 = 60
\][/tex]
First, simplify the left-hand side of the equation. We'll start with the term [tex]\(-x(x-2)\)[/tex]:
[tex]\[
-x(x-2) = -x^2 + 2x
\][/tex]
Substituting this back into the equation, we have:
[tex]\[
-x^2 + 2x + x + (2x + 8) + 10 = 60
\][/tex]
Now, combine all the like terms on the left-hand side:
[tex]\[
-x^2 + 5x + 8 + 10 = 60
\][/tex]
Simplify further by combining the constants:
[tex]\[
-x^2 + 5x + 18 = 60
\][/tex]
To solve for [tex]\(x\)[/tex], we need to set the equation to zero. Subtract 60 from both sides:
[tex]\[
-x^2 + 5x + 18 - 60 = 0
\][/tex]
Simplify the constants:
[tex]\[
-x^2 + 5x - 42 = 0
\][/tex]
Now, to solve the quadratic equation [tex]\(-x^2 + 5x - 42 = 0\)[/tex], we can use the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In our equation, [tex]\(a = -1\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = -42\)[/tex]. Substituting these values into the quadratic formula:
[tex]\[
x = \frac{-5 \pm \sqrt{5^2 - 4(-1)(-42)}}{2(-1)}
\][/tex]
Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = 5^2 - 4(-1)(-42) = 25 - 168 = -143
\][/tex]
Since the discriminant is negative, we will have complex solutions. Computing the square root of [tex]\(-143\)[/tex]:
[tex]\[
\sqrt{-143} = i\sqrt{143}
\][/tex]
Substitute back into the quadratic formula:
[tex]\[
x = \frac{-5 \pm \sqrt{-143}}{-2} = \frac{-5 \pm i\sqrt{143}}{-2}
\][/tex]
Simplify:
[tex]\[
x = \frac{-5}{-2} \pm \frac{i\sqrt{143}}{-2} = \frac{5}{2} \mp \frac{i\sqrt{143}}{2}
\][/tex]
Thus, the solutions to the equation are:
[tex]\[
x = \frac{5}{2} - \frac{\sqrt{143}i}{2} \quad \text{and} \quad x = \frac{5}{2} + \frac{\sqrt{143}i}{2}
\][/tex]
So the solutions are:
[tex]\[
x = \frac{5}{2} - \frac{\sqrt{143}i}{2}, \quad x = \frac{5}{2} + \frac{\sqrt{143}i}{2}
\][/tex]
