Finite Math Examples

$2 x^{2} + 4 x + 2 = 10$

Step 1

Subtract $10$ from both sides of the equation.

$2 x^{2} + 4 x + 2 - 10 = 0$

Step 2

Subtract $10$ from $2$ .

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

Step 3

Factor $2$ out of $2 x^{2} + 4 x - 8$ .

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Step 3.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 4 x - 8 = 0$

Step 3.2

Factor $2$ out of $4 x$ .

$2 (x^{2}) + 2 (2 x) - 8 = 0$

Step 3.3

Factor $2$ out of $- 8$ .

$2 x^{2} + 2 (2 x) + 2 \cdot - 4 = 0$

Step 3.4

Factor $2$ out of $2 x^{2} + 2 (2 x)$ .

$2 (x^{2} + 2 x) + 2 \cdot - 4 = 0$

Step 3.5

Factor $2$ out of $2 (x^{2} + 2 x) + 2 \cdot - 4$ .

$2 (x^{2} + 2 x - 4) = 0$

Step 4

Divide each term in $2 (x^{2} + 2 x - 4) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (x^{2} + 2 x - 4) = 0$ by $2$ .

$\frac{2 (x^{2} + 2 x - 4)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 2 x - 4)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $x^{2} + 2 x - 4$ by $1$ .

$x^{2} + 2 x - 4 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$x^{2} + 2 x - 4 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = 2$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 2 \pm \sqrt{4 + 16}}{2 \cdot 1}$

Step 7.1.3

Add $4$ and $16$ .

$x = \frac{- 2 \pm \sqrt{20}}{2 \cdot 1}$

Step 7.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 7.1.4.1

Factor $4$ out of $20$ .

$x = \frac{- 2 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 7.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{5}}{2}$

Step 7.3

Simplify $\frac{- 2 \pm 2 \sqrt{5}}{2}$ .

$x = - 1 \pm \sqrt{5}$

Step 8

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{5}, - 1 - \sqrt{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - 1 + \sqrt{5}, - 1 - \sqrt{5}$

Decimal Form:

$x = 1.23606797 \dots, - 3.23606797 \dots$