Calculus Examples

$90 = 7 x^{2} + 2 x h + 2 (7 x h)$

Step 1

Rewrite the equation as $7 x^{2} + 2 x h + 2 \cdot (7 x h) = 90$ .

$7 x^{2} + 2 x h + 2 \cdot (7 x h) = 90$

Step 2

Simplify $7 x^{2} + 2 x h + 2 \cdot (7 x h)$ .

Tap for more steps...

Step 2.1

Multiply $7$ by $2$ .

$7 x^{2} + 2 x h + 14 x h = 90$

Step 2.2

Add $2 x h$ and $14 x h$ .

$7 x^{2} + 16 x h = 90$

Step 3

Subtract $90$ from both sides of the equation.

$7 x^{2} + 16 x h - 90 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 7$ , $b = 16 h$ , and $c = - 90$ into the quadratic formula and solve for $x$ .

$\frac{- 16 h \pm \sqrt{{(16 h)}^{2} - 4 \cdot (7 \cdot - 90)}}{2 \cdot 7}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Apply the product rule to $16 h$ .

$x = \frac{- 16 h \pm \sqrt{16^{2} h^{2} - 4 \cdot 7 \cdot - 90}}{2 \cdot 7}$

Step 6.1.2

Raise $16$ to the power of $2$ .

$x = \frac{- 16 h \pm \sqrt{256 h^{2} - 4 \cdot 7 \cdot - 90}}{2 \cdot 7}$

Step 6.1.3

Multiply $- 4 \cdot 7 \cdot - 90$ .

Tap for more steps...

Step 6.1.3.1

Multiply $- 4$ by $7$ .

$x = \frac{- 16 h \pm \sqrt{256 h^{2} - 28 \cdot - 90}}{2 \cdot 7}$

Step 6.1.3.2

Multiply $- 28$ by $- 90$ .

$x = \frac{- 16 h \pm \sqrt{256 h^{2} + 2520}}{2 \cdot 7}$

Step 6.1.4

Factor $8$ out of $256 h^{2} + 2520$ .

Tap for more steps...

Step 6.1.4.1

Factor $8$ out of $256 h^{2}$ .

$x = \frac{- 16 h \pm \sqrt{8 (32 h^{2}) + 2520}}{2 \cdot 7}$

Step 6.1.4.2

Factor $8$ out of $2520$ .

$x = \frac{- 16 h \pm \sqrt{8 (32 h^{2}) + 8 (315)}}{2 \cdot 7}$

Step 6.1.4.3

Factor $8$ out of $8 (32 h^{2}) + 8 (315)$ .

$x = \frac{- 16 h \pm \sqrt{8 (32 h^{2} + 315)}}{2 \cdot 7}$

Step 6.1.5

Rewrite $8 (32 h^{2} + 315)$ as $2^{2} \cdot (2 (32 h^{2} + 315))$ .

Tap for more steps...

Step 6.1.5.1

Factor $4$ out of $8$ .

$x = \frac{- 16 h \pm \sqrt{4 (2) (32 h^{2} + 315)}}{2 \cdot 7}$

Step 6.1.5.2

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

$x = \frac{- 16 h \pm \sqrt{2^{2} \cdot (2 (32 h^{2} + 315))}}{2 \cdot 7}$

Step 6.1.5.3

Add parentheses.

$x = \frac{- 16 h \pm \sqrt{2^{2} \cdot (2 (32 h^{2} + 315))}}{2 \cdot 7}$

Step 6.1.6

Pull terms out from under the radical.

$x = \frac{- 16 h \pm 2 \sqrt{2 (32 h^{2} + 315)}}{2 \cdot 7}$

Step 6.2

Multiply $2$ by $7$ .

$x = \frac{- 16 h \pm 2 \sqrt{2 (32 h^{2} + 315)}}{14}$

Step 6.3

Simplify $\frac{- 16 h \pm 2 \sqrt{2 (32 h^{2} + 315)}}{14}$ .

$x = \frac{- 8 h \pm \sqrt{2 (32 h^{2} + 315)}}{7}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{8 h - \sqrt{2 (32 h^{2} + 315)}}{7}$

$x = - \frac{8 h + \sqrt{2 (32 h^{2} + 315)}}{7}$