Solve for y 20y^2-20y-5=y

$20 y^{2} - 20 y - 5 = y$

Step 1

Move all terms containing $y$ to the left side of the equation.

Subtract $y$ from both sides of the equation.

$20 y^{2} - 20 y - 5 - y = 0$

Subtract $y$ from $- 20 y$ .

$20 y^{2} - 21 y - 5 = 0$

Step 2

Factor by grouping.

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 $a \cdot c = 20 \cdot - 5 = - 100$ and whose sum is $b = - 21$ .

Factor $- 21$ out of $- 21 y$ .

$20 y^{2} - 21 y - 5 = 0$

Rewrite $- 21$ as $4$ plus $- 25$

$20 y^{2} + (4 - 25) y - 5 = 0$

Apply the distributive property.

$20 y^{2} + 4 y - 25 y - 5 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(20 y^{2} + 4 y) - 25 y - 5 = 0$

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

$4 y (5 y + 1) - 5 (5 y + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 y + 1$ .

$(5 y + 1) (4 y - 5) = 0$

Step 3

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

$5 y + 1 = 0$

$4 y - 5 = 0$

Step 4

Set $5 y + 1$ equal to $0$ and solve for $y$ .

Set $5 y + 1$ equal to $0$ .

$5 y + 1 = 0$

Solve $5 y + 1 = 0$ for $y$ .

Subtract $1$ from both sides of the equation.

$5 y = - 1$

Divide each term in $5 y = - 1$ by $5$ and simplify.

Divide each term in $5 y = - 1$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 5

Set $4 y - 5$ equal to $0$ and solve for $y$ .

Set $4 y - 5$ equal to $0$ .

$4 y - 5 = 0$

Solve $4 y - 5 = 0$ for $y$ .

Add $5$ to both sides of the equation.

$4 y = 5$

Divide each term in $4 y = 5$ by $4$ and simplify.

Divide each term in $4 y = 5$ by $4$ .

$\frac{4 y}{4} = \frac{5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{5}{4}$

Divide $y$ by $1$ .

$y = \frac{5}{4}$

Step 6

The final solution is all the values that make $(5 y + 1) (4 y - 5) = 0$ true.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = - 0.2, 1.25$

Mixed Number Form:

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

Solve for y 20y^2-20y-5=y

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