Calculus Examples

$x^{2} + 9 x - 4 = x + 2$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $x$ from both sides of the equation.

$x^{2} + 9 x - 4 - x = 2$

Step 1.2

Subtract $x$ from $9 x$ .

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

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the equation.

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

Step 2.2

Subtract $2$ from $- 4$ .

$x^{2} + 8 x - 6 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = 8$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot - 6)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 6}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{- 8 \pm \sqrt{64 + 24}}{2 \cdot 1}$

Step 5.1.3

Add $64$ and $24$ .

$x = \frac{- 8 \pm \sqrt{88}}{2 \cdot 1}$

Step 5.1.4

Rewrite $88$ as $2^{2} \cdot 22$ .

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Step 5.1.4.1

Factor $4$ out of $88$ .

$x = \frac{- 8 \pm \sqrt{4 (22)}}{2 \cdot 1}$

Step 5.1.4.2

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

$x = \frac{- 8 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 2 \sqrt{22}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 2 \sqrt{22}}{2}$

Step 5.3

Simplify $\frac{- 8 \pm 2 \sqrt{22}}{2}$ .

$x = - 4 \pm \sqrt{22}$

Step 6

The final answer is the combination of both solutions.

$x = - 4 + \sqrt{22}, - 4 - \sqrt{22}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 4 + \sqrt{22}, - 4 - \sqrt{22}$

Decimal Form:

$x = 0.69041575 \dots, - 8.69041575 \dots$