Precalculus Examples

$2 y^{2} - y - \frac{1}{2} = 0$

Step 1

To write $2 y^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- y + 2 y^{2} \cdot \frac{2}{2} - \frac{1}{2} = 0$

Step 2

Combine $2 y^{2}$ and $\frac{2}{2}$ .

$- y + \frac{2 y^{2} \cdot 2}{2} - \frac{1}{2} = 0$

Step 3

Combine the numerators over the common denominator.

$- y + \frac{2 y^{2} \cdot 2 - 1}{2} = 0$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Multiply $2$ by $2$ .

$- y + \frac{4 y^{2} - 1}{2} = 0$

Step 4.2

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$- y + \frac{{(2 y)}^{2} - 1}{2} = 0$

Step 4.3

Rewrite $1$ as $1^{2}$ .

$- y + \frac{{(2 y)}^{2} - 1^{2}}{2} = 0$

Step 4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 y$ and $b = 1$ .

$- y + \frac{(2 y + 1) (2 y - 1)}{2} = 0$

Step 5

To write $- y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- y \cdot \frac{2}{2} + \frac{(2 y + 1) (2 y - 1)}{2} = 0$

Step 6

Combine $- y$ and $\frac{2}{2}$ .

$\frac{- y \cdot 2}{2} + \frac{(2 y + 1) (2 y - 1)}{2} = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{- y \cdot 2 + (2 y + 1) (2 y - 1)}{2} = 0$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply $2$ by $- 1$ .

$\frac{- 2 y + (2 y + 1) (2 y - 1)}{2} = 0$

Step 8.2

Expand $(2 y + 1) (2 y - 1)$ using the FOIL Method.

Tap for more steps...

Step 8.2.1

Apply the distributive property.

$\frac{- 2 y + 2 y (2 y - 1) + 1 (2 y - 1)}{2} = 0$

Step 8.2.2

Apply the distributive property.

$\frac{- 2 y + 2 y (2 y) + 2 y \cdot - 1 + 1 (2 y - 1)}{2} = 0$

Step 8.2.3

Apply the distributive property.

$\frac{- 2 y + 2 y (2 y) + 2 y \cdot - 1 + 1 (2 y) + 1 \cdot - 1}{2} = 0$

Step 8.3

Combine the opposite terms in $2 y (2 y) + 2 y \cdot - 1 + 1 (2 y) + 1 \cdot - 1$ .

Tap for more steps...

Step 8.3.1

Reorder the factors in the terms $2 y \cdot - 1$ and $1 (2 y)$ .

$\frac{- 2 y + 2 y (2 y) - 1 \cdot (2 y) + 1 \cdot (2 y) + 1 \cdot - 1}{2} = 0$

Step 8.3.2

Add $- 1 \cdot 2 y$ and $1 \cdot 2 y$ .

$\frac{- 2 y + 2 y (2 y) + 0 + 1 \cdot - 1}{2} = 0$

Step 8.3.3

Add $2 y (2 y)$ and $0$ .

$\frac{- 2 y + 2 y (2 y) + 1 \cdot - 1}{2} = 0$

Step 8.4

Simplify each term.

Tap for more steps...

Step 8.4.1

Rewrite using the commutative property of multiplication.

$\frac{- 2 y + 2 \cdot (2 y \cdot y) + 1 \cdot - 1}{2} = 0$

Step 8.4.2

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 8.4.2.1

Move $y$ .

$\frac{- 2 y + 2 \cdot (2 (y \cdot y)) + 1 \cdot - 1}{2} = 0$

Step 8.4.2.2

Multiply $y$ by $y$ .

$\frac{- 2 y + 2 \cdot (2 y^{2}) + 1 \cdot - 1}{2} = 0$

Step 8.4.3

Multiply $2$ by $2$ .

$\frac{- 2 y + 4 y^{2} + 1 \cdot - 1}{2} = 0$

Step 8.4.4

Multiply $- 1$ by $1$ .

$\frac{- 2 y + 4 y^{2} - 1}{2} = 0$

Step 8.5

Reorder terms.

$\frac{4 y^{2} - 2 y - 1}{2} = 0$

Step 9

Set the numerator equal to zero.

$4 y^{2} - 2 y - 1 = 0$

Step 10

Solve the equation for $y$ .

Tap for more steps...

Step 10.1

Use the quadratic formula to find the solutions.

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

Step 10.2

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

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

Step 10.3

Simplify.

Tap for more steps...

Step 10.3.1

Simplify the numerator.

Tap for more steps...

Step 10.3.1.1

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

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

Step 10.3.1.2

Multiply $- 4 \cdot 4 \cdot - 1$ .

Tap for more steps...

Step 10.3.1.2.1

Multiply $- 4$ by $4$ .

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

Step 10.3.1.2.2

Multiply $- 16$ by $- 1$ .

$y = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 4}$

Step 10.3.1.3

Add $4$ and $16$ .

$y = \frac{2 \pm \sqrt{20}}{2 \cdot 4}$

Step 10.3.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 10.3.1.4.1

Factor $4$ out of $20$ .

$y = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 4}$

Step 10.3.1.4.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 4}$

Step 10.3.1.5

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 4}$

Step 10.3.2

Multiply $2$ by $4$ .

$y = \frac{2 \pm 2 \sqrt{5}}{8}$

Step 10.3.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{8}$ .

$y = \frac{1 \pm \sqrt{5}}{4}$

Step 10.4

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

Tap for more steps...

Step 10.4.1

Simplify the numerator.

Tap for more steps...

Step 10.4.1.1

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

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

Step 10.4.1.2

Multiply $- 4 \cdot 4 \cdot - 1$ .

Tap for more steps...

Step 10.4.1.2.1

Multiply $- 4$ by $4$ .

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

Step 10.4.1.2.2

Multiply $- 16$ by $- 1$ .

$y = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 4}$

Step 10.4.1.3

Add $4$ and $16$ .

$y = \frac{2 \pm \sqrt{20}}{2 \cdot 4}$

Step 10.4.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 10.4.1.4.1

Factor $4$ out of $20$ .

$y = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 4}$

Step 10.4.1.4.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 4}$

Step 10.4.1.5

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 4}$

Step 10.4.2

Multiply $2$ by $4$ .

$y = \frac{2 \pm 2 \sqrt{5}}{8}$

Step 10.4.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{8}$ .

$y = \frac{1 \pm \sqrt{5}}{4}$

Step 10.4.4

Change the $\pm$ to $+$ .

$y = \frac{1 + \sqrt{5}}{4}$

Step 10.5

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

Tap for more steps...

Step 10.5.1

Simplify the numerator.

Tap for more steps...

Step 10.5.1.1

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

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

Step 10.5.1.2

Multiply $- 4 \cdot 4 \cdot - 1$ .

Tap for more steps...

Step 10.5.1.2.1

Multiply $- 4$ by $4$ .

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

Step 10.5.1.2.2

Multiply $- 16$ by $- 1$ .

$y = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 4}$

Step 10.5.1.3

Add $4$ and $16$ .

$y = \frac{2 \pm \sqrt{20}}{2 \cdot 4}$

Step 10.5.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 10.5.1.4.1

Factor $4$ out of $20$ .

$y = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 4}$

Step 10.5.1.4.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 4}$

Step 10.5.1.5

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 4}$

Step 10.5.2

Multiply $2$ by $4$ .

$y = \frac{2 \pm 2 \sqrt{5}}{8}$

Step 10.5.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{8}$ .

$y = \frac{1 \pm \sqrt{5}}{4}$

Step 10.5.4

Change the $\pm$ to $-$ .

$y = \frac{1 - \sqrt{5}}{4}$

Step 10.6

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{5}}{4}, \frac{1 - \sqrt{5}}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$y = \frac{1 + \sqrt{5}}{4}, \frac{1 - \sqrt{5}}{4}$

Decimal Form:

$y = 0.80901699 \dots, - 0.30901699 \dots$