Pre-Algebra Examples

$9 y = 5 y^{2} + 4$

Step 1

Subtract $5 y^{2}$ from both sides of the equation.

$9 y - 5 y^{2} = 4$

Step 2

Subtract $4$ from both sides of the equation.

$9 y - 5 y^{2} - 4 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $9 y - 5 y^{2} - 4$ .

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Step 3.1.1

Reorder $9 y$ and $- 5 y^{2}$ .

$- 5 y^{2} + 9 y - 4 = 0$

Step 3.1.2

Factor $- 1$ out of $- 5 y^{2}$ .

$- (5 y^{2}) + 9 y - 4 = 0$

Step 3.1.3

Factor $- 1$ out of $9 y$ .

$- (5 y^{2}) - (- 9 y) - 4 = 0$

Step 3.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$- (5 y^{2}) - (- 9 y) - 1 \cdot 4 = 0$

Step 3.1.5

Factor $- 1$ out of $- (5 y^{2}) - (- 9 y)$ .

$- (5 y^{2} - 9 y) - 1 \cdot 4 = 0$

Step 3.1.6

Factor $- 1$ out of $- (5 y^{2} - 9 y) - 1 (4)$ .

$- (5 y^{2} - 9 y + 4) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor by grouping.

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Step 3.2.1.1

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 $a \cdot c = 5 \cdot 4 = 20$ and whose sum is $b = - 9$ .

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Step 3.2.1.1.1

Factor $- 9$ out of $- 9 y$ .

$- (5 y^{2} - 9 y + 4) = 0$

Step 3.2.1.1.2

Rewrite $- 9$ as $- 4$ plus $- 5$

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

Step 3.2.1.1.3

Apply the distributive property.

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

Step 3.2.1.2

Factor out the greatest common factor from each group.

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Step 3.2.1.2.1

Group the first two terms and the last two terms.

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

Step 3.2.1.2.2

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

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

Step 3.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 y - 4$ .

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

Step 3.2.2

Remove unnecessary parentheses.

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

Step 4

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

$5 y - 4 = 0$

$y - 1 = 0$

Step 5

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

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Step 5.1

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

$5 y - 4 = 0$

Step 5.2

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

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Step 5.2.1

Add $4$ to both sides of the equation.

$5 y = 4$

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 6

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$y = 1$

Step 7

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

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