Algebra Examples

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

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 $a = 1$ , $b = - 8$ , and $c = x - 5$ into the quadratic formula and solve for $y$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot (x - 5))}}{2 \cdot 1}$

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

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot (x - 5)}}{2 \cdot 1}$

Step 1.3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (x - 5)}}{2 \cdot 1}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{8 \pm \sqrt{64 - 4 x - 4 \cdot - 5}}{2 \cdot 1}$

Step 1.3.1.4

Multiply $- 4$ by $- 5$ .

$y = \frac{8 \pm \sqrt{64 - 4 x + 20}}{2 \cdot 1}$

Step 1.3.1.5

Add $64$ and $20$ .

$y = \frac{8 \pm \sqrt{- 4 x + 84}}{2 \cdot 1}$

Step 1.3.1.6

Factor $4$ out of $- 4 x + 84$ .

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Step 1.3.1.6.1

Factor $4$ out of $- 4 x$ .

$y = \frac{8 \pm \sqrt{4 (- x) + 84}}{2 \cdot 1}$

Step 1.3.1.6.2

Factor $4$ out of $84$ .

$y = \frac{8 \pm \sqrt{4 (- x) + 4 (21)}}{2 \cdot 1}$

Step 1.3.1.6.3

Factor $4$ out of $4 (- x) + 4 (21)$ .

$y = \frac{8 \pm \sqrt{4 (- x + 21)}}{2 \cdot 1}$

Step 1.3.1.7

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

$y = \frac{8 \pm \sqrt{2^{2} (- x + 21)}}{2 \cdot 1}$

Step 1.3.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm 2 \sqrt{- x + 21}}{2 \cdot 1}$

Step 1.3.2

Multiply $2$ by $1$ .

$y = \frac{8 \pm 2 \sqrt{- x + 21}}{2}$

Step 1.3.3

Simplify $\frac{8 \pm 2 \sqrt{- x + 21}}{2}$ .

$y = 4 \pm \sqrt{- x + 21}$

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

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot (x - 5)}}{2 \cdot 1}$

Step 1.4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (x - 5)}}{2 \cdot 1}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{8 \pm \sqrt{64 - 4 x - 4 \cdot - 5}}{2 \cdot 1}$

Step 1.4.1.4

Multiply $- 4$ by $- 5$ .

$y = \frac{8 \pm \sqrt{64 - 4 x + 20}}{2 \cdot 1}$

Step 1.4.1.5

Add $64$ and $20$ .

$y = \frac{8 \pm \sqrt{- 4 x + 84}}{2 \cdot 1}$

Step 1.4.1.6

Factor $4$ out of $- 4 x + 84$ .

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Step 1.4.1.6.1

Factor $4$ out of $- 4 x$ .

$y = \frac{8 \pm \sqrt{4 (- x) + 84}}{2 \cdot 1}$

Step 1.4.1.6.2

Factor $4$ out of $84$ .

$y = \frac{8 \pm \sqrt{4 (- x) + 4 (21)}}{2 \cdot 1}$

Step 1.4.1.6.3

Factor $4$ out of $4 (- x) + 4 (21)$ .

$y = \frac{8 \pm \sqrt{4 (- x + 21)}}{2 \cdot 1}$

Step 1.4.1.7

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

$y = \frac{8 \pm \sqrt{2^{2} (- x + 21)}}{2 \cdot 1}$

Step 1.4.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm 2 \sqrt{- x + 21}}{2 \cdot 1}$

Step 1.4.2

Multiply $2$ by $1$ .

$y = \frac{8 \pm 2 \sqrt{- x + 21}}{2}$

Step 1.4.3

Simplify $\frac{8 \pm 2 \sqrt{- x + 21}}{2}$ .

$y = 4 \pm \sqrt{- x + 21}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = 4 + \sqrt{- x + 21}$

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

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot (x - 5)}}{2 \cdot 1}$

Step 1.5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (x - 5)}}{2 \cdot 1}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{8 \pm \sqrt{64 - 4 x - 4 \cdot - 5}}{2 \cdot 1}$

Step 1.5.1.4

Multiply $- 4$ by $- 5$ .

$y = \frac{8 \pm \sqrt{64 - 4 x + 20}}{2 \cdot 1}$

Step 1.5.1.5

Add $64$ and $20$ .

$y = \frac{8 \pm \sqrt{- 4 x + 84}}{2 \cdot 1}$

Step 1.5.1.6

Factor $4$ out of $- 4 x + 84$ .

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Step 1.5.1.6.1

Factor $4$ out of $- 4 x$ .

$y = \frac{8 \pm \sqrt{4 (- x) + 84}}{2 \cdot 1}$

Step 1.5.1.6.2

Factor $4$ out of $84$ .

$y = \frac{8 \pm \sqrt{4 (- x) + 4 (21)}}{2 \cdot 1}$

Step 1.5.1.6.3

Factor $4$ out of $4 (- x) + 4 (21)$ .

$y = \frac{8 \pm \sqrt{4 (- x + 21)}}{2 \cdot 1}$

Step 1.5.1.7

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

$y = \frac{8 \pm \sqrt{2^{2} (- x + 21)}}{2 \cdot 1}$

Step 1.5.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm 2 \sqrt{- x + 21}}{2 \cdot 1}$

Step 1.5.2

Multiply $2$ by $1$ .

$y = \frac{8 \pm 2 \sqrt{- x + 21}}{2}$

Step 1.5.3

Simplify $\frac{8 \pm 2 \sqrt{- x + 21}}{2}$ .

$y = 4 \pm \sqrt{- x + 21}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = 4 - \sqrt{- x + 21}$

Step 1.6

The final answer is the combination of both solutions.

$y = 4 + \sqrt{- x + 21}$

$y = 4 - \sqrt{- x + 21}$

$y = 4 + \sqrt{- x + 21}$

$y = 4 - \sqrt{- x + 21}$

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 $4 + \sqrt{- x + 21}, 4 - \sqrt{- x + 21}$ .

$y = 4 + \sqrt{- x + 21}$

$y = 4 - \sqrt{- x + 21}$

Step 4