Basic Math Examples

$r = 2 y^{2} + 2 y - 1$

Step 1

Rewrite the equation as $2 y^{2} + 2 y - 1 = r$ .

$2 y^{2} + 2 y - 1 = r$

Step 2

Subtract $r$ from both sides of the equation.

$2 y^{2} + 2 y - 1 - r = 0$

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 $a = 2$ , $b = 2$ , and $c = - 1 - r$ into the quadratic formula and solve for $y$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $2$ out of $2^{2} - 4 \cdot 2 \cdot (- 1 - r)$ .

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Step 5.1.1.1

Factor $2$ out of $2^{2}$ .

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

Step 5.1.1.2

Factor $2$ out of $- 4 \cdot 2 \cdot (- 1 - r)$ .

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

Step 5.1.1.3

Factor $2$ out of $2 \cdot 2 + 2 (- 2 \cdot 2 \cdot (- 1 - r))$ .

$y = \frac{- 2 \pm \sqrt{2 (2 - 2 \cdot 2 \cdot (- 1 - r))}}{2 \cdot 2}$

Step 5.1.2

Factor $2$ out of $2 - 2 \cdot 2 \cdot (- 1 - r)$ .

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Step 5.1.2.1

Factor $2$ out of $2$ .

$y = \frac{- 2 \pm \sqrt{2 (2 (1) - 2 \cdot 2 \cdot (- 1 - r))}}{2 \cdot 2}$

Step 5.1.2.2

Factor $2$ out of $- 2 \cdot 2 \cdot (- 1 - r)$ .

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

Step 5.1.2.3

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 2 \cdot (- 1 - r))$ .

$y = \frac{- 2 \pm \sqrt{2 (2 (1 - 1 \cdot 2 \cdot (- 1 - r)))}}{2 \cdot 2}$

Step 5.1.3

Multiply $- 1$ by $2$ .

$y = \frac{- 2 \pm \sqrt{2 (2 (1 - 2 \cdot (- 1 - r)))}}{2 \cdot 2}$

Step 5.1.4

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{2 (2 (1 - 2 \cdot - 1 - 2 (- r)))}}{2 \cdot 2}$

Step 5.1.5

Multiply $- 2$ by $- 1$ .

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

Step 5.1.6

Multiply $- 1$ by $- 2$ .

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

Step 5.1.7

Add $1$ and $2$ .

$y = \frac{- 2 \pm \sqrt{2 \cdot (2 (3 + 2 r))}}{2 \cdot 2}$

Step 5.1.8

Multiply $2$ by $2$ .

$y = \frac{- 2 \pm \sqrt{4 (3 + 2 r)}}{2 \cdot 2}$

Step 5.1.9

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

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

Step 5.1.10

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{3 + 2 r}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

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

Step 5.3

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

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

Step 6

The final answer is the combination of both solutions.

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

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