Solve for k k^2+7k-5=4

$k^{2} + 7 k - 5 = 4$

Step 1

Move all terms to the left side of the equation and simplify.

Subtract $4$ from both sides of the equation.

$k^{2} + 7 k - 5 - 4 = 0$

Subtract $4$ from $- 5$ .

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

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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

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

Step 4

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$k = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 9$ .

Multiply $- 4$ by $1$ .

$k = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 9}}{2 \cdot 1}$

Multiply $- 4$ by $- 9$ .

$k = \frac{- 7 \pm \sqrt{49 + 36}}{2 \cdot 1}$

Add $49$ and $36$ .

$k = \frac{- 7 \pm \sqrt{85}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$k = \frac{- 7 \pm \sqrt{85}}{2}$

Step 5

The final answer is the combination of both solutions.

$k = - \frac{7 - \sqrt{85}}{2}, - \frac{7 + \sqrt{85}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$k = - \frac{7 - \sqrt{85}}{2}, - \frac{7 + \sqrt{85}}{2}$

Decimal Form:

$k = 1.10977222 \dots, - 8.10977222 \dots$