Algebra Examples

$- 5 x^{2} - x + 56 = x + 15$

Step 1

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

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

Subtract $x$ from both sides of the equation.

$- 5 x^{2} - x + 56 - x = 15$

Step 1.1.2

Subtract $15$ from both sides of the equation.

$- 5 x^{2} - x + 56 - x - 15 = 0$

Step 1.2

Simplify $- 5 x^{2} - x + 56 - x - 15$ .

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Step 1.2.1

Subtract $x$ from $- x$ .

$- 5 x^{2} - 2 x + 56 - 15 = 0$

Step 1.2.2

Subtract $15$ from $56$ .

$- 5 x^{2} - 2 x + 41 = 0$

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 $a = - 5$ , $b = - 2$ , and $c = 41$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (- 5 \cdot 41)}}{2 \cdot - 5}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 5 \cdot 41}}{2 \cdot - 5}$

Step 4.1.2

Multiply $- 4 \cdot - 5 \cdot 41$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 20 \cdot 41}}{2 \cdot - 5}$

Step 4.1.2.2

Multiply $20$ by $41$ .

$x = \frac{2 \pm \sqrt{4 + 820}}{2 \cdot - 5}$

Step 4.1.3

Add $4$ and $820$ .

$x = \frac{2 \pm \sqrt{824}}{2 \cdot - 5}$

Step 4.1.4

Rewrite $824$ as $2^{2} \cdot 206$ .

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Step 4.1.4.1

Factor $4$ out of $824$ .

$x = \frac{2 \pm \sqrt{4 (206)}}{2 \cdot - 5}$

Step 4.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 206}}{2 \cdot - 5}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{206}}{2 \cdot - 5}$

Step 4.2

Multiply $2$ by $- 5$ .

$x = \frac{2 \pm 2 \sqrt{206}}{- 10}$

Step 4.3

Simplify $\frac{2 \pm 2 \sqrt{206}}{- 10}$ .

$x = \frac{1 \pm \sqrt{206}}{- 5}$

Step 4.4

Move the negative in front of the fraction.

$x = - \frac{1 \pm \sqrt{206}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 \pm \sqrt{206}}{5}$

Decimal Form:

$x = - 3.07054001 \dots, 2.67054001 \dots$