Basic Math Examples

$15 y^{2} - 2 = y$

Step 1

Subtract $y$ from both sides of the equation.

$15 y^{2} - 2 - y = 0$

Step 2

Factor by grouping.

Tap for more steps...

Step 2.1

Reorder terms.

$15 y^{2} - y - 2 = 0$

Step 2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 15 \cdot - 2 = - 30$ and whose sum is $b = - 1$ .

Tap for more steps...

Step 2.2.1

Factor $- 1$ out of $- y$ .

$15 y^{2} - (y) - 2 = 0$

Step 2.2.2

Rewrite $- 1$ as $5$ plus $- 6$

$15 y^{2} + (5 - 6) y - 2 = 0$

Step 2.2.3

Apply the distributive property.

$15 y^{2} + 5 y - 6 y - 2 = 0$

Step 2.3

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.3.1

Group the first two terms and the last two terms.

$(15 y^{2} + 5 y) - 6 y - 2 = 0$

Step 2.3.2

Factor out the greatest common factor (GCF) from each group.

$5 y (3 y + 1) - 2 (3 y + 1) = 0$

Step 2.4

Factor the polynomial by factoring out the greatest common factor, $3 y + 1$ .

$(3 y + 1) (5 y - 2) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 y + 1 = 0$

$5 y - 2 = 0$

Step 4

Set $3 y + 1$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 4.1

Set $3 y + 1$ equal to $0$ .

$3 y + 1 = 0$

Step 4.2

Solve $3 y + 1 = 0$ for $y$ .

Tap for more steps...

Step 4.2.1

Subtract $1$ from both sides of the equation.

$3 y = - 1$

Step 4.2.2

Divide each term in $3 y = - 1$ by $3$ and simplify.

Tap for more steps...

Step 4.2.2.1

Divide each term in $3 y = - 1$ by $3$ .

$\frac{3 y}{3} = \frac{- 1}{3}$

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- 1}{3}$

Step 4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 1}{3}$

Step 4.2.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.2.3.1

Move the negative in front of the fraction.

$y = - \frac{1}{3}$

Step 5

Set $5 y - 2$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 5.1

Set $5 y - 2$ equal to $0$ .

$5 y - 2 = 0$

Step 5.2

Solve $5 y - 2 = 0$ for $y$ .

Tap for more steps...

Step 5.2.1

Add $2$ to both sides of the equation.

$5 y = 2$

Step 5.2.2

Divide each term in $5 y = 2$ by $5$ and simplify.

Tap for more steps...

Step 5.2.2.1

Divide each term in $5 y = 2$ by $5$ .

$\frac{5 y}{5} = \frac{2}{5}$

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{2}{5}$

Step 5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{5}$

Step 6

The final solution is all the values that make $(3 y + 1) (5 y - 2) = 0$ true.

$y = - \frac{1}{3}, \frac{2}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{1}{3}, \frac{2}{5}$

Decimal Form:

$y = - 0.\overline{3}, 0.4$