Pre-Algebra Examples

$x = 4 y^{2} + 8 y - 3$

Step 1

Rewrite the equation as $4 y^{2} + 8 y - 3 = x$ .

$4 y^{2} + 8 y - 3 = x$

Step 2

Subtract $x$ from both sides of the equation.

$4 y^{2} + 8 y - 3 - x = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = 8$ , and $c = - 3 - x$ into the quadratic formula and solve for $y$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (4 \cdot (- 3 - x))}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot 4 \cdot (- 3 - x)}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4$ by $4$ .

$y = \frac{- 8 \pm \sqrt{64 - 16 \cdot (- 3 - x)}}{2 \cdot 4}$

Step 5.1.3

Apply the distributive property.

$y = \frac{- 8 \pm \sqrt{64 - 16 \cdot - 3 - 16 (- x)}}{2 \cdot 4}$

Step 5.1.4

Multiply $- 16$ by $- 3$ .

$y = \frac{- 8 \pm \sqrt{64 + 48 - 16 (- x)}}{2 \cdot 4}$

Step 5.1.5

Multiply $- 1$ by $- 16$ .

$y = \frac{- 8 \pm \sqrt{64 + 48 + 16 x}}{2 \cdot 4}$

Step 5.1.6

Add $64$ and $48$ .

$y = \frac{- 8 \pm \sqrt{112 + 16 x}}{2 \cdot 4}$

Step 5.1.7

Factor $16$ out of $112 + 16 x$ .

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Step 5.1.7.1

Factor $16$ out of $112$ .

$y = \frac{- 8 \pm \sqrt{16 \cdot 7 + 16 x}}{2 \cdot 4}$

Step 5.1.7.2

Factor $16$ out of $16 \cdot 7 + 16 x$ .

$y = \frac{- 8 \pm \sqrt{16 (7 + x)}}{2 \cdot 4}$

Step 5.1.8

Rewrite $16 (7 + x)$ as ${(2^{2})}^{2} (7 + x)$ .

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Step 5.1.8.1

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

$y = \frac{- 8 \pm \sqrt{4^{2} (7 + x)}}{2 \cdot 4}$

Step 5.1.8.2

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

$y = \frac{- 8 \pm \sqrt{{(2^{2})}^{2} (7 + x)}}{2 \cdot 4}$

Step 5.1.9

Pull terms out from under the radical.

$y = \frac{- 8 \pm 2^{2} \sqrt{7 + x}}{2 \cdot 4}$

Step 5.1.10

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

$y = \frac{- 8 \pm 4 \sqrt{7 + x}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$y = \frac{- 8 \pm 4 \sqrt{7 + x}}{8}$

Step 5.3

Simplify $\frac{- 8 \pm 4 \sqrt{7 + x}}{8}$ .

$y = \frac{- 2 \pm \sqrt{7 + x}}{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot 4 \cdot (- 3 - x)}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4$ by $4$ .

$y = \frac{- 8 \pm \sqrt{64 - 16 \cdot (- 3 - x)}}{2 \cdot 4}$

Step 6.1.3

Apply the distributive property.

$y = \frac{- 8 \pm \sqrt{64 - 16 \cdot - 3 - 16 (- x)}}{2 \cdot 4}$

Step 6.1.4

Multiply $- 16$ by $- 3$ .

$y = \frac{- 8 \pm \sqrt{64 + 48 - 16 (- x)}}{2 \cdot 4}$

Step 6.1.5

Multiply $- 1$ by $- 16$ .

$y = \frac{- 8 \pm \sqrt{64 + 48 + 16 x}}{2 \cdot 4}$

Step 6.1.6

Add $64$ and $48$ .

$y = \frac{- 8 \pm \sqrt{112 + 16 x}}{2 \cdot 4}$

Step 6.1.7

Factor $16$ out of $112 + 16 x$ .

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Step 6.1.7.1

Factor $16$ out of $112$ .

$y = \frac{- 8 \pm \sqrt{16 \cdot 7 + 16 x}}{2 \cdot 4}$

Step 6.1.7.2

Factor $16$ out of $16 \cdot 7 + 16 x$ .

$y = \frac{- 8 \pm \sqrt{16 (7 + x)}}{2 \cdot 4}$

Step 6.1.8

Rewrite $16 (7 + x)$ as ${(2^{2})}^{2} (7 + x)$ .

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Step 6.1.8.1

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

$y = \frac{- 8 \pm \sqrt{4^{2} (7 + x)}}{2 \cdot 4}$

Step 6.1.8.2

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

$y = \frac{- 8 \pm \sqrt{{(2^{2})}^{2} (7 + x)}}{2 \cdot 4}$

Step 6.1.9

Pull terms out from under the radical.

$y = \frac{- 8 \pm 2^{2} \sqrt{7 + x}}{2 \cdot 4}$

Step 6.1.10

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

$y = \frac{- 8 \pm 4 \sqrt{7 + x}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$y = \frac{- 8 \pm 4 \sqrt{7 + x}}{8}$

Step 6.3

Simplify $\frac{- 8 \pm 4 \sqrt{7 + x}}{8}$ .

$y = \frac{- 2 \pm \sqrt{7 + x}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$y = \frac{- 2 + \sqrt{7 + x}}{2}$

Step 6.5

Rewrite $- 2$ as $- 1 (2)$ .

$y = \frac{- 1 \cdot 2 + \sqrt{7 + x}}{2}$

Step 6.6

Factor $- 1$ out of $\sqrt{7 + x}$ .

$y = \frac{- 1 \cdot 2 - 1 (- \sqrt{7 + x})}{2}$

Step 6.7

Factor $- 1$ out of $- 1 (2) - 1 (- \sqrt{7 + x})$ .

$y = \frac{- 1 (2 - \sqrt{7 + x})}{2}$

Step 6.8

Move the negative in front of the fraction.

$y = - \frac{2 - \sqrt{7 + x}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot 4 \cdot (- 3 - x)}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4$ by $4$ .

$y = \frac{- 8 \pm \sqrt{64 - 16 \cdot (- 3 - x)}}{2 \cdot 4}$

Step 7.1.3

Apply the distributive property.

$y = \frac{- 8 \pm \sqrt{64 - 16 \cdot - 3 - 16 (- x)}}{2 \cdot 4}$

Step 7.1.4

Multiply $- 16$ by $- 3$ .

$y = \frac{- 8 \pm \sqrt{64 + 48 - 16 (- x)}}{2 \cdot 4}$

Step 7.1.5

Multiply $- 1$ by $- 16$ .

$y = \frac{- 8 \pm \sqrt{64 + 48 + 16 x}}{2 \cdot 4}$

Step 7.1.6

Add $64$ and $48$ .

$y = \frac{- 8 \pm \sqrt{112 + 16 x}}{2 \cdot 4}$

Step 7.1.7

Factor $16$ out of $112 + 16 x$ .

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Step 7.1.7.1

Factor $16$ out of $112$ .

$y = \frac{- 8 \pm \sqrt{16 \cdot 7 + 16 x}}{2 \cdot 4}$

Step 7.1.7.2

Factor $16$ out of $16 \cdot 7 + 16 x$ .

$y = \frac{- 8 \pm \sqrt{16 (7 + x)}}{2 \cdot 4}$

Step 7.1.8

Rewrite $16 (7 + x)$ as ${(2^{2})}^{2} (7 + x)$ .

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Step 7.1.8.1

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

$y = \frac{- 8 \pm \sqrt{4^{2} (7 + x)}}{2 \cdot 4}$

Step 7.1.8.2

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

$y = \frac{- 8 \pm \sqrt{{(2^{2})}^{2} (7 + x)}}{2 \cdot 4}$

Step 7.1.9

Pull terms out from under the radical.

$y = \frac{- 8 \pm 2^{2} \sqrt{7 + x}}{2 \cdot 4}$

Step 7.1.10

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

$y = \frac{- 8 \pm 4 \sqrt{7 + x}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$y = \frac{- 8 \pm 4 \sqrt{7 + x}}{8}$

Step 7.3

Simplify $\frac{- 8 \pm 4 \sqrt{7 + x}}{8}$ .

$y = \frac{- 2 \pm \sqrt{7 + x}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$y = \frac{- 2 - \sqrt{7 + x}}{2}$

Step 7.5

Factor $- 1$ out of $- 2 - \sqrt{7 + x}$ .

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Step 7.5.1

Reorder $7$ and $x$ .

$y = \frac{- 2 - \sqrt{x + 7}}{2}$

Step 7.5.2

Rewrite $- 2$ as $- 1 (2)$ .

$y = \frac{- 1 \cdot 2 - \sqrt{x + 7}}{2}$

Step 7.5.3

Factor $- 1$ out of $- \sqrt{x + 7}$ .

$y = \frac{- 1 \cdot 2 - (\sqrt{x + 7})}{2}$

Step 7.5.4

Factor $- 1$ out of $- 1 (2) - (\sqrt{x + 7})$ .

$y = \frac{- 1 (2 + \sqrt{x + 7})}{2}$

Step 7.5.5

Rewrite $- 1 (2 + \sqrt{x + 7})$ as $- (2 + \sqrt{x + 7})$ .

$y = \frac{- (2 + \sqrt{x + 7})}{2}$

Step 7.6

Move the negative in front of the fraction.

$y = - \frac{2 + \sqrt{x + 7}}{2}$

Step 8

The final answer is the combination of both solutions.

$y = - \frac{2 - \sqrt{7 + x}}{2}$

$y = - \frac{2 + \sqrt{x + 7}}{2}$

Step 9

The given equation $y = - \frac{2 - \sqrt{7 + x}}{2}, - \frac{2 + \sqrt{x + 7}}{2}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$