Precalculus Examples

$5 y^{2} + 40 y + 80 x - 400 = 0$

Step 1

Solve for $y$ .

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 5$ , $b = 40$ , and $c = 80 x - 400$ into the quadratic formula and solve for $y$ .

$\frac{- 40 \pm \sqrt{40^{2} - 4 \cdot (5 \cdot (80 x - 400))}}{2 \cdot 5}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Raise $40$ to the power of $2$ .

$y = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 5 \cdot (80 x - 400)}}{2 \cdot 5}$

Step 1.3.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 40 \pm \sqrt{1600 - 20 \cdot (80 x - 400)}}{2 \cdot 5}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{- 40 \pm \sqrt{1600 - 20 (80 x) - 20 \cdot - 400}}{2 \cdot 5}$

Step 1.3.1.4

Multiply $80$ by $- 20$ .

$y = \frac{- 40 \pm \sqrt{1600 - 1600 x - 20 \cdot - 400}}{2 \cdot 5}$

Step 1.3.1.5

Multiply $- 20$ by $- 400$ .

$y = \frac{- 40 \pm \sqrt{1600 - 1600 x + 8000}}{2 \cdot 5}$

Step 1.3.1.6

Add $1600$ and $8000$ .

$y = \frac{- 40 \pm \sqrt{- 1600 x + 9600}}{2 \cdot 5}$

Step 1.3.1.7

Factor $1600$ out of $- 1600 x + 9600$ .

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Step 1.3.1.7.1

Factor $1600$ out of $- 1600 x$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x) + 9600}}{2 \cdot 5}$

Step 1.3.1.7.2

Factor $1600$ out of $9600$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x) + 1600 (6)}}{2 \cdot 5}$

Step 1.3.1.7.3

Factor $1600$ out of $1600 (- x) + 1600 (6)$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x + 6)}}{2 \cdot 5}$

Step 1.3.1.8

Rewrite $1600$ as $40^{2}$ .

$y = \frac{- 40 \pm \sqrt{40^{2} (- x + 6)}}{2 \cdot 5}$

Step 1.3.1.9

Pull terms out from under the radical.

$y = \frac{- 40 \pm 40 \sqrt{- x + 6}}{2 \cdot 5}$

Step 1.3.2

Multiply $2$ by $5$ .

$y = \frac{- 40 \pm 40 \sqrt{- x + 6}}{10}$

Step 1.3.3

Simplify $\frac{- 40 \pm 40 \sqrt{- x + 6}}{10}$ .

$y = - 4 \pm 4 \sqrt{- x + 6}$

Step 1.4

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

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $40$ to the power of $2$ .

$y = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 5 \cdot (80 x - 400)}}{2 \cdot 5}$

Step 1.4.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 40 \pm \sqrt{1600 - 20 \cdot (80 x - 400)}}{2 \cdot 5}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 40 \pm \sqrt{1600 - 20 (80 x) - 20 \cdot - 400}}{2 \cdot 5}$

Step 1.4.1.4

Multiply $80$ by $- 20$ .

$y = \frac{- 40 \pm \sqrt{1600 - 1600 x - 20 \cdot - 400}}{2 \cdot 5}$

Step 1.4.1.5

Multiply $- 20$ by $- 400$ .

$y = \frac{- 40 \pm \sqrt{1600 - 1600 x + 8000}}{2 \cdot 5}$

Step 1.4.1.6

Add $1600$ and $8000$ .

$y = \frac{- 40 \pm \sqrt{- 1600 x + 9600}}{2 \cdot 5}$

Step 1.4.1.7

Factor $1600$ out of $- 1600 x + 9600$ .

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Step 1.4.1.7.1

Factor $1600$ out of $- 1600 x$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x) + 9600}}{2 \cdot 5}$

Step 1.4.1.7.2

Factor $1600$ out of $9600$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x) + 1600 (6)}}{2 \cdot 5}$

Step 1.4.1.7.3

Factor $1600$ out of $1600 (- x) + 1600 (6)$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x + 6)}}{2 \cdot 5}$

Step 1.4.1.8

Rewrite $1600$ as $40^{2}$ .

$y = \frac{- 40 \pm \sqrt{40^{2} (- x + 6)}}{2 \cdot 5}$

Step 1.4.1.9

Pull terms out from under the radical.

$y = \frac{- 40 \pm 40 \sqrt{- x + 6}}{2 \cdot 5}$

Step 1.4.2

Multiply $2$ by $5$ .

$y = \frac{- 40 \pm 40 \sqrt{- x + 6}}{10}$

Step 1.4.3

Simplify $\frac{- 40 \pm 40 \sqrt{- x + 6}}{10}$ .

$y = - 4 \pm 4 \sqrt{- x + 6}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = - 4 + 4 \sqrt{- x + 6}$

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $40$ to the power of $2$ .

$y = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 5 \cdot (80 x - 400)}}{2 \cdot 5}$

Step 1.5.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 40 \pm \sqrt{1600 - 20 \cdot (80 x - 400)}}{2 \cdot 5}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 40 \pm \sqrt{1600 - 20 (80 x) - 20 \cdot - 400}}{2 \cdot 5}$

Step 1.5.1.4

Multiply $80$ by $- 20$ .

$y = \frac{- 40 \pm \sqrt{1600 - 1600 x - 20 \cdot - 400}}{2 \cdot 5}$

Step 1.5.1.5

Multiply $- 20$ by $- 400$ .

$y = \frac{- 40 \pm \sqrt{1600 - 1600 x + 8000}}{2 \cdot 5}$

Step 1.5.1.6

Add $1600$ and $8000$ .

$y = \frac{- 40 \pm \sqrt{- 1600 x + 9600}}{2 \cdot 5}$

Step 1.5.1.7

Factor $1600$ out of $- 1600 x + 9600$ .

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Step 1.5.1.7.1

Factor $1600$ out of $- 1600 x$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x) + 9600}}{2 \cdot 5}$

Step 1.5.1.7.2

Factor $1600$ out of $9600$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x) + 1600 (6)}}{2 \cdot 5}$

Step 1.5.1.7.3

Factor $1600$ out of $1600 (- x) + 1600 (6)$ .

$y = \frac{- 40 \pm \sqrt{1600 (- x + 6)}}{2 \cdot 5}$

Step 1.5.1.8

Rewrite $1600$ as $40^{2}$ .

$y = \frac{- 40 \pm \sqrt{40^{2} (- x + 6)}}{2 \cdot 5}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{- 40 \pm 40 \sqrt{- x + 6}}{2 \cdot 5}$

Step 1.5.2

Multiply $2$ by $5$ .

$y = \frac{- 40 \pm 40 \sqrt{- x + 6}}{10}$

Step 1.5.3

Simplify $\frac{- 40 \pm 40 \sqrt{- x + 6}}{10}$ .

$y = - 4 \pm 4 \sqrt{- x + 6}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = - 4 - 4 \sqrt{- x + 6}$

Step 1.6

The final answer is the combination of both solutions.

$y = - 4 + 4 \sqrt{- x + 6}$

$y = - 4 - 4 \sqrt{- x + 6}$

$y = - 4 + 4 \sqrt{- x + 6}$

$y = - 4 - 4 \sqrt{- x + 6}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

The standard form is $- 4 + 4 \sqrt{- x + 6}, - 4 - 4 \sqrt{- x + 6}$ .

$y = - 4 + 4 \sqrt{- x + 6}$

$y = - 4 - 4 \sqrt{- x + 6}$

Step 4