Calculus Examples

$f (x) = 5 x^{2} + 7$ , $f (x) = - 3 x + 8$

Add $- 3 x$ to both sides of the equation.

$5 x^{2} + 7 + 3 x = 8$

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

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Move $8$ to the left side of the equation by subtracting it from both sides.

$5 x^{2} + 7 + 3 x - 8 = 0$

Subtract $8$ from $7$ .

$5 x^{2} + 3 x - 1 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 5$ , $b = 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (5 \cdot - 1)}}{2 \cdot 5}$

Simplify.

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (5 \cdot - 1)}}{2 \cdot 5}$

Multiply $5$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 5}$

Multiply $- 4$ by $- 5$ .

$x = \frac{- 3 \pm \sqrt{9 + 20}}{2 \cdot 5}$

Add $9$ and $20$ .

$x = \frac{- 3 \pm \sqrt{29}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 3 \pm \sqrt{29}}{10}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (5 \cdot - 1)}}{2 \cdot 5}$

Multiply $5$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 5}$

Multiply $- 4$ by $- 5$ .

$x = \frac{- 3 \pm \sqrt{9 + 20}}{2 \cdot 5}$

Add $9$ and $20$ .

$x = \frac{- 3 \pm \sqrt{29}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 3 \pm \sqrt{29}}{10}$

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{29}}{10}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{29}}{10}$

Factor $- 1$ out of $\sqrt{29}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{29})}{10}$

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{29})$ .

$x = \frac{- 1 (3 - \sqrt{29})}{10}$

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{29}}{10}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (5 \cdot - 1)}}{2 \cdot 5}$

Multiply $5$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 5}$

Multiply $- 4$ by $- 5$ .

$x = \frac{- 3 \pm \sqrt{9 + 20}}{2 \cdot 5}$

Add $9$ and $20$ .

$x = \frac{- 3 \pm \sqrt{29}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 3 \pm \sqrt{29}}{10}$

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{29}}{10}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{29}}{10}$

Factor $- 1$ out of $- \sqrt{29}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{29})}{10}$

Factor $- 1$ out of $- 1 (3) - (\sqrt{29})$ .

$x = \frac{- 1 (3 + \sqrt{29})}{10}$

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{29}}{10}$

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{29}}{10}, - \frac{3 + \sqrt{29}}{10}$

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