Algebra Examples

$4 k (k + 2) = 2$

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $4 k (k + 2)$ .

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Step 1.1.1.1

Apply the distributive property.

$4 k \cdot k + 4 k \cdot 2 = 2$

Step 1.1.1.2

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

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Step 1.1.1.2.1

Move $k$ .

$4 (k \cdot k) + 4 k \cdot 2 = 2$

Step 1.1.1.2.2

Multiply $k$ by $k$ .

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

Step 1.1.1.3

Multiply $2$ by $4$ .

$4 k^{2} + 8 k = 2$

Step 1.2

Subtract $2$ from both sides of the equation.

$4 k^{2} + 8 k - 2 = 0$

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 $a = 4$ , $b = 8$ , and $c = - 2$ into the quadratic formula and solve for $k$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (4 \cdot - 2)}}{2 \cdot 4}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $8$ to the power of $2$ .

$k = \frac{- 8 \pm \sqrt{64 - 4 \cdot 4 \cdot - 2}}{2 \cdot 4}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $4$ .

$k = \frac{- 8 \pm \sqrt{64 - 16 \cdot - 2}}{2 \cdot 4}$

Step 4.1.2.2

Multiply $- 16$ by $- 2$ .

$k = \frac{- 8 \pm \sqrt{64 + 32}}{2 \cdot 4}$

Step 4.1.3

Add $64$ and $32$ .

$k = \frac{- 8 \pm \sqrt{96}}{2 \cdot 4}$

Step 4.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 4.1.4.1

Factor $16$ out of $96$ .

$k = \frac{- 8 \pm \sqrt{16 (6)}}{2 \cdot 4}$

Step 4.1.4.2

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

$k = \frac{- 8 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 4}$

Step 4.1.5

Pull terms out from under the radical.

$k = \frac{- 8 \pm 4 \sqrt{6}}{2 \cdot 4}$

Step 4.2

Multiply $2$ by $4$ .

$k = \frac{- 8 \pm 4 \sqrt{6}}{8}$

Step 4.3

Simplify $\frac{- 8 \pm 4 \sqrt{6}}{8}$ .

$k = \frac{- 2 \pm \sqrt{6}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$k = \frac{- 2 \pm \sqrt{6}}{2}$

Decimal Form:

$k = 0.22474487 \dots, - 2.22474487 \dots$