Solve Using the Quadratic Formula 4k(k+5)=5

$4 k (k + 5) = 5$

Step 1

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

Simplify the left side.

Simplify $4 k (k + 5)$ .

Apply the distributive property.

$4 k \cdot k + 4 k \cdot 5 = 5$

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

Move $k$ .

$4 (k \cdot k) + 4 k \cdot 5 = 5$

Multiply $k$ by $k$ .

$4 k^{2} + 4 k \cdot 5 = 5$

Multiply $5$ by $4$ .

$4 k^{2} + 20 k = 5$

Subtract $5$ from both sides of the equation.

$4 k^{2} + 20 k - 5 = 0$

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 $a = 4$ , $b = 20$ , and $c = - 5$ into the quadratic formula and solve for $k$ .

$\frac{- 20 \pm \sqrt{20^{2} - 4 \cdot (4 \cdot - 5)}}{2 \cdot 4}$

Step 4

Simplify.

Simplify the numerator.

Raise $20$ to the power of $2$ .

$k = \frac{- 20 \pm \sqrt{400 - 4 \cdot 4 \cdot - 5}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 5$ .

Multiply $- 4$ by $4$ .

$k = \frac{- 20 \pm \sqrt{400 - 16 \cdot - 5}}{2 \cdot 4}$

Multiply $- 16$ by $- 5$ .

$k = \frac{- 20 \pm \sqrt{400 + 80}}{2 \cdot 4}$

Add $400$ and $80$ .

$k = \frac{- 20 \pm \sqrt{480}}{2 \cdot 4}$

Rewrite $480$ as $4^{2} \cdot 30$ .

Factor $16$ out of $480$ .

$k = \frac{- 20 \pm \sqrt{16 (30)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$k = \frac{- 20 \pm \sqrt{4^{2} \cdot 30}}{2 \cdot 4}$

Pull terms out from under the radical.

$k = \frac{- 20 \pm 4 \sqrt{30}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$k = \frac{- 20 \pm 4 \sqrt{30}}{8}$

Simplify $\frac{- 20 \pm 4 \sqrt{30}}{8}$ .

$k = \frac{- 5 \pm \sqrt{30}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$k = \frac{- 5 \pm \sqrt{30}}{2}$

Decimal Form:

$k = 0.23861278 \dots, - 5.23861278 \dots$

Solve Using the Quadratic Formula 4k(k+5)=5

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