Algebra Examples

$4 x - y^{2} = 2 y + 13$

Step 1

Solve for $y$ .

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Step 1.1

Subtract $2 y$ from both sides of the equation.

$4 x - y^{2} - 2 y = 13$

Step 1.2

Subtract $13$ from both sides of the equation.

$4 x - y^{2} - 2 y - 13 = 0$

Step 1.3

Use the quadratic formula to find the solutions.

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

Step 1.4

Substitute the values $a = - 1$ , $b = - 2$ , and $c = 4 x - 13$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (- 1 \cdot (4 x - 13))}}{2 \cdot - 1}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (4 x - 13)}}{2 \cdot - 1}$

Step 1.5.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{2 \pm \sqrt{4 + 4 \cdot (4 x - 13)}}{2 \cdot - 1}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 + 4 (4 x) + 4 \cdot - 13}}{2 \cdot - 1}$

Step 1.5.1.4

Multiply $4$ by $4$ .

$y = \frac{2 \pm \sqrt{4 + 16 x + 4 \cdot - 13}}{2 \cdot - 1}$

Step 1.5.1.5

Multiply $4$ by $- 13$ .

$y = \frac{2 \pm \sqrt{4 + 16 x - 52}}{2 \cdot - 1}$

Step 1.5.1.6

Subtract $52$ from $4$ .

$y = \frac{2 \pm \sqrt{16 x - 48}}{2 \cdot - 1}$

Step 1.5.1.7

Factor $16$ out of $16 x - 48$ .

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Step 1.5.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{2 \pm \sqrt{16 (x) - 48}}{2 \cdot - 1}$

Step 1.5.1.7.2

Factor $16$ out of $- 48$ .

$y = \frac{2 \pm \sqrt{16 x + 16 \cdot - 3}}{2 \cdot - 1}$

Step 1.5.1.7.3

Factor $16$ out of $16 x + 16 \cdot - 3$ .

$y = \frac{2 \pm \sqrt{16 (x - 3)}}{2 \cdot - 1}$

Step 1.5.1.8

Rewrite $16 (x - 3)$ as ${(2^{2})}^{2} (x - 3)$ .

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Step 1.5.1.8.1

Rewrite $16$ as $4^{2}$ .

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

Step 1.5.1.8.2

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

$y = \frac{2 \pm \sqrt{{(2^{2})}^{2} (x - 3)}}{2 \cdot - 1}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{2 \pm 2^{2} \sqrt{x - 3}}{2 \cdot - 1}$

Step 1.5.1.10

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

$y = \frac{2 \pm 4 \sqrt{x - 3}}{2 \cdot - 1}$

Step 1.5.2

Multiply $2$ by $- 1$ .

$y = \frac{2 \pm 4 \sqrt{x - 3}}{- 2}$

Step 1.5.3

Simplify $\frac{2 \pm 4 \sqrt{x - 3}}{- 2}$ .

$y = \frac{1 \pm 2 \sqrt{x - 3}}{- 1}$

Step 1.5.4

Move the negative one from the denominator of $\frac{1 \pm 2 \sqrt{x - 3}}{- 1}$ .

$y = - 1 \cdot (1 \pm 2 \sqrt{x - 3})$

Step 1.5.5

Rewrite $- 1 \cdot (1 \pm 2 \sqrt{x - 3})$ as $- (1 \pm 2 \sqrt{x - 3})$ .

$y = - (1 \pm 2 \sqrt{x - 3})$

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (4 x - 13)}}{2 \cdot - 1}$

Step 1.6.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{2 \pm \sqrt{4 + 4 \cdot (4 x - 13)}}{2 \cdot - 1}$

Step 1.6.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 + 4 (4 x) + 4 \cdot - 13}}{2 \cdot - 1}$

Step 1.6.1.4

Multiply $4$ by $4$ .

$y = \frac{2 \pm \sqrt{4 + 16 x + 4 \cdot - 13}}{2 \cdot - 1}$

Step 1.6.1.5

Multiply $4$ by $- 13$ .

$y = \frac{2 \pm \sqrt{4 + 16 x - 52}}{2 \cdot - 1}$

Step 1.6.1.6

Subtract $52$ from $4$ .

$y = \frac{2 \pm \sqrt{16 x - 48}}{2 \cdot - 1}$

Step 1.6.1.7

Factor $16$ out of $16 x - 48$ .

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Step 1.6.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{2 \pm \sqrt{16 (x) - 48}}{2 \cdot - 1}$

Step 1.6.1.7.2

Factor $16$ out of $- 48$ .

$y = \frac{2 \pm \sqrt{16 x + 16 \cdot - 3}}{2 \cdot - 1}$

Step 1.6.1.7.3

Factor $16$ out of $16 x + 16 \cdot - 3$ .

$y = \frac{2 \pm \sqrt{16 (x - 3)}}{2 \cdot - 1}$

Step 1.6.1.8

Rewrite $16 (x - 3)$ as ${(2^{2})}^{2} (x - 3)$ .

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Step 1.6.1.8.1

Rewrite $16$ as $4^{2}$ .

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

Step 1.6.1.8.2

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

$y = \frac{2 \pm \sqrt{{(2^{2})}^{2} (x - 3)}}{2 \cdot - 1}$

Step 1.6.1.9

Pull terms out from under the radical.

$y = \frac{2 \pm 2^{2} \sqrt{x - 3}}{2 \cdot - 1}$

Step 1.6.1.10

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

$y = \frac{2 \pm 4 \sqrt{x - 3}}{2 \cdot - 1}$

Step 1.6.2

Multiply $2$ by $- 1$ .

$y = \frac{2 \pm 4 \sqrt{x - 3}}{- 2}$

Step 1.6.3

Simplify $\frac{2 \pm 4 \sqrt{x - 3}}{- 2}$ .

$y = \frac{1 \pm 2 \sqrt{x - 3}}{- 1}$

Step 1.6.4

Move the negative one from the denominator of $\frac{1 \pm 2 \sqrt{x - 3}}{- 1}$ .

$y = - 1 \cdot (1 \pm 2 \sqrt{x - 3})$

Step 1.6.5

Rewrite $- 1 \cdot (1 \pm 2 \sqrt{x - 3})$ as $- (1 \pm 2 \sqrt{x - 3})$ .

$y = - (1 \pm 2 \sqrt{x - 3})$

Step 1.6.6

Change the $\pm$ to $+$ .

$y = - (1 + 2 \sqrt{x - 3})$

Step 1.6.7

Apply the distributive property.

$y = - 1 \cdot 1 - (2 \sqrt{x - 3})$

Step 1.6.8

Multiply $- 1$ by $1$ .

$y = - 1 - (2 \sqrt{x - 3})$

Step 1.6.9

Multiply $2$ by $- 1$ .

$y = - 1 - 2 \sqrt{x - 3}$

Step 1.7

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

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Step 1.7.1

Simplify the numerator.

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Step 1.7.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (4 x - 13)}}{2 \cdot - 1}$

Step 1.7.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{2 \pm \sqrt{4 + 4 \cdot (4 x - 13)}}{2 \cdot - 1}$

Step 1.7.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 + 4 (4 x) + 4 \cdot - 13}}{2 \cdot - 1}$

Step 1.7.1.4

Multiply $4$ by $4$ .

$y = \frac{2 \pm \sqrt{4 + 16 x + 4 \cdot - 13}}{2 \cdot - 1}$

Step 1.7.1.5

Multiply $4$ by $- 13$ .

$y = \frac{2 \pm \sqrt{4 + 16 x - 52}}{2 \cdot - 1}$

Step 1.7.1.6

Subtract $52$ from $4$ .

$y = \frac{2 \pm \sqrt{16 x - 48}}{2 \cdot - 1}$

Step 1.7.1.7

Factor $16$ out of $16 x - 48$ .

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Step 1.7.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{2 \pm \sqrt{16 (x) - 48}}{2 \cdot - 1}$

Step 1.7.1.7.2

Factor $16$ out of $- 48$ .

$y = \frac{2 \pm \sqrt{16 x + 16 \cdot - 3}}{2 \cdot - 1}$

Step 1.7.1.7.3

Factor $16$ out of $16 x + 16 \cdot - 3$ .

$y = \frac{2 \pm \sqrt{16 (x - 3)}}{2 \cdot - 1}$

Step 1.7.1.8

Rewrite $16 (x - 3)$ as ${(2^{2})}^{2} (x - 3)$ .

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Step 1.7.1.8.1

Rewrite $16$ as $4^{2}$ .

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

Step 1.7.1.8.2

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

$y = \frac{2 \pm \sqrt{{(2^{2})}^{2} (x - 3)}}{2 \cdot - 1}$

Step 1.7.1.9

Pull terms out from under the radical.

$y = \frac{2 \pm 2^{2} \sqrt{x - 3}}{2 \cdot - 1}$

Step 1.7.1.10

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

$y = \frac{2 \pm 4 \sqrt{x - 3}}{2 \cdot - 1}$

Step 1.7.2

Multiply $2$ by $- 1$ .

$y = \frac{2 \pm 4 \sqrt{x - 3}}{- 2}$

Step 1.7.3

Simplify $\frac{2 \pm 4 \sqrt{x - 3}}{- 2}$ .

$y = \frac{1 \pm 2 \sqrt{x - 3}}{- 1}$

Step 1.7.4

Move the negative one from the denominator of $\frac{1 \pm 2 \sqrt{x - 3}}{- 1}$ .

$y = - 1 \cdot (1 \pm 2 \sqrt{x - 3})$

Step 1.7.5

Rewrite $- 1 \cdot (1 \pm 2 \sqrt{x - 3})$ as $- (1 \pm 2 \sqrt{x - 3})$ .

$y = - (1 \pm 2 \sqrt{x - 3})$

Step 1.7.6

Change the $\pm$ to $-$ .

$y = - (1 - 2 \sqrt{x - 3})$

Step 1.7.7

Apply the distributive property.

$y = - 1 \cdot 1 - (- 2 \sqrt{x - 3})$

Step 1.7.8

Multiply $- 1$ by $1$ .

$y = - 1 - (- 2 \sqrt{x - 3})$

Step 1.7.9

Multiply $- 2$ by $- 1$ .

$y = - 1 + 2 \sqrt{x - 3}$

Step 1.8

The final answer is the combination of both solutions.

$y = - 1 - 2 \sqrt{x - 3}$

$y = - 1 + 2 \sqrt{x - 3}$

$y = - 1 - 2 \sqrt{x - 3}$

$y = - 1 + 2 \sqrt{x - 3}$

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 $- 1 - 2 \sqrt{x - 3}, - 1 + 2 \sqrt{x - 3}$ .

$y = - 1 - 2 \sqrt{x - 3}$

$y = - 1 + 2 \sqrt{x - 3}$

Step 4