Find All Complex Solutions -x^2+9x-7=0

$- x^{2} + 9 x - 7 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = - 1$ , $b = 9$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (- 1 \cdot - 7)}}{2 \cdot - 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 1 \cdot - 7}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 7$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 9 \pm \sqrt{81 + 4 \cdot - 7}}{2 \cdot - 1}$

Multiply $4$ by $- 7$ .

$x = \frac{- 9 \pm \sqrt{81 - 28}}{2 \cdot - 1}$

Subtract $28$ from $81$ .

$x = \frac{- 9 \pm \sqrt{53}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 9 \pm \sqrt{53}}{- 2}$

Simplify $\frac{- 9 \pm \sqrt{53}}{- 2}$ .

$x = \frac{9 \pm \sqrt{53}}{2}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{53}}{2}, \frac{9 - \sqrt{53}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9 + \sqrt{53}}{2}, \frac{9 - \sqrt{53}}{2}$

Decimal Form:

$x = 8.14005494 \dots, 0.85994505 \dots$