Trigonometry Examples

$p (3800000) = x^{2} - 4000 x + 7800000$

Step 1

Rewrite the equation as $x^{2} - 4000 x + 7800000 = p (3800000)$ .

$x^{2} - 4000 x + 7800000 = p (3800000)$

Step 2

Move $3800000$ to the left of $p$ .

$x^{2} - 4000 x + 7800000 = 3800000 p$

Step 3

Subtract $3800000 p$ from both sides of the equation.

$x^{2} - 4000 x + 7800000 - 3800000 p = 0$

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 $a = 1$ , $b = - 4000$ , and $c = 7800000 - 3800000 p$ into the quadratic formula and solve for $x$ .

$\frac{4000 \pm \sqrt{{(- 4000)}^{2} - 4 \cdot (1 \cdot (7800000 - 3800000 p))}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 4000$ to the power of $2$ .

$x = \frac{4000 \pm \sqrt{16000000 - 4 \cdot 1 \cdot (7800000 - 3800000 p)}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4$ by $1$ .

$x = \frac{4000 \pm \sqrt{16000000 - 4 \cdot (7800000 - 3800000 p)}}{2 \cdot 1}$

Step 6.1.3

Apply the distributive property.

$x = \frac{4000 \pm \sqrt{16000000 - 4 \cdot 7800000 - 4 (- 3800000 p)}}{2 \cdot 1}$

Step 6.1.4

Multiply $- 4$ by $7800000$ .

$x = \frac{4000 \pm \sqrt{16000000 - 31200000 - 4 (- 3800000 p)}}{2 \cdot 1}$

Step 6.1.5

Multiply $- 3800000$ by $- 4$ .

$x = \frac{4000 \pm \sqrt{16000000 - 31200000 + 15200000 p}}{2 \cdot 1}$

Step 6.1.6

Subtract $31200000$ from $16000000$ .

$x = \frac{4000 \pm \sqrt{- 15200000 + 15200000 p}}{2 \cdot 1}$

Step 6.1.7

Factor $15200000$ out of $- 15200000 + 15200000 p$ .

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Step 6.1.7.1

Factor $15200000$ out of $- 15200000$ .

$x = \frac{4000 \pm \sqrt{15200000 \cdot - 1 + 15200000 p}}{2 \cdot 1}$

Step 6.1.7.2

Factor $15200000$ out of $15200000 \cdot - 1 + 15200000 p$ .

$x = \frac{4000 \pm \sqrt{15200000 (- 1 + p)}}{2 \cdot 1}$

Step 6.1.8

Rewrite $15200000 (- 1 + p)$ as ${(20^{2})}^{2} \cdot (95 (- 1 + p))$ .

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Step 6.1.8.1

Factor $160000$ out of $15200000$ .

$x = \frac{4000 \pm \sqrt{160000 (95) (- 1 + p)}}{2 \cdot 1}$

Step 6.1.8.2

Rewrite $160000$ as $400^{2}$ .

$x = \frac{4000 \pm \sqrt{400^{2} \cdot (95 (- 1 + p))}}{2 \cdot 1}$

Step 6.1.8.3

Rewrite $400$ as $20^{2}$ .

$x = \frac{4000 \pm \sqrt{{(20^{2})}^{2} \cdot (95 (- 1 + p))}}{2 \cdot 1}$

Step 6.1.8.4

Add parentheses.

$x = \frac{4000 \pm \sqrt{{(20^{2})}^{2} \cdot (95 (- 1 + p))}}{2 \cdot 1}$

Step 6.1.9

Pull terms out from under the radical.

$x = \frac{4000 \pm 20^{2} \sqrt{95 (- 1 + p)}}{2 \cdot 1}$

Step 6.1.10

Raise $20$ to the power of $2$ .

$x = \frac{4000 \pm 400 \sqrt{95 (- 1 + p)}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{4000 \pm 400 \sqrt{95 (- 1 + p)}}{2}$

Step 6.3

Simplify $\frac{4000 \pm 400 \sqrt{95 (- 1 + p)}}{2}$ .

$x = 2000 \pm 200 \sqrt{95 (- 1 + p)}$

Step 7

The final answer is the combination of both solutions.

$x = 2000 + 200 \sqrt{95 (- 1 + p)}$

$x = 2000 - 200 \sqrt{95 (- 1 + p)}$