Solve Using the Quadratic Formula 2k(k+11)=11

$2 k (k + 11) = 11$

Step 1

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

Simplify the left side.

Simplify $2 k (k + 11)$ .

Apply the distributive property.

$2 k \cdot k + 2 k \cdot 11 = 11$

Multiply $k$ by $k$ by adding the exponents.

Move $k$ .

$2 (k \cdot k) + 2 k \cdot 11 = 11$

Multiply $k$ by $k$ .

$2 k^{2} + 2 k \cdot 11 = 11$

Multiply $11$ by $2$ .

$2 k^{2} + 22 k = 11$

Subtract $11$ from both sides of the equation.

$2 k^{2} + 22 k - 11 = 0$

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 $a = 2$ , $b = 22$ , and $c = - 11$ into the quadratic formula and solve for $k$ .

$\frac{- 22 \pm \sqrt{22^{2} - 4 \cdot (2 \cdot - 11)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $22$ to the power of $2$ .

$k = \frac{- 22 \pm \sqrt{484 - 4 \cdot 2 \cdot - 11}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 11$ .

Multiply $- 4$ by $2$ .

$k = \frac{- 22 \pm \sqrt{484 - 8 \cdot - 11}}{2 \cdot 2}$

Multiply $- 8$ by $- 11$ .

$k = \frac{- 22 \pm \sqrt{484 + 88}}{2 \cdot 2}$

Add $484$ and $88$ .

$k = \frac{- 22 \pm \sqrt{572}}{2 \cdot 2}$

Rewrite $572$ as $2^{2} \cdot 143$ .

Factor $4$ out of $572$ .

$k = \frac{- 22 \pm \sqrt{4 (143)}}{2 \cdot 2}$

Rewrite $4$ as $2^{2}$ .

$k = \frac{- 22 \pm \sqrt{2^{2} \cdot 143}}{2 \cdot 2}$

Pull terms out from under the radical.

$k = \frac{- 22 \pm 2 \sqrt{143}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$k = \frac{- 22 \pm 2 \sqrt{143}}{4}$

Simplify $\frac{- 22 \pm 2 \sqrt{143}}{4}$ .

$k = \frac{- 11 \pm \sqrt{143}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$k = \frac{- 11 \pm \sqrt{143}}{2}$

Decimal Form:

$k = 0.47913037 \dots, - 11.47913037 \dots$

Solve Using the Quadratic Formula 2k(k+11)=11

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