Basic Math Examples

$5 y^{2} - 4 y - 1 = 0$

Step 1

Factor by grouping.

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Step 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 $a \cdot c = 5 \cdot - 1 = - 5$ and whose sum is $b = - 4$ .

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Step 1.1.1

Factor $- 4$ out of $- 4 y$ .

$5 y^{2} - 4 y - 1 = 0$

Step 1.1.2

Rewrite $- 4$ as $1$ plus $- 5$

$5 y^{2} + (1 - 5) y - 1 = 0$

Step 1.1.3

Apply the distributive property.

$5 y^{2} + 1 y - 5 y - 1 = 0$

Step 1.1.4

Multiply $y$ by $1$ .

$5 y^{2} + y - 5 y - 1 = 0$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$(5 y^{2} + y) - 5 y - 1 = 0$

Step 1.2.2

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

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

Step 1.3

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

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

Step 2

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

$5 y + 1 = 0$

$y - 1 = 0$

Step 3

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

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Step 3.1

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

$5 y + 1 = 0$

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$5 y = - 1$

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

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

Step 4

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

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Step 4.1

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

$y - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$y = 1$

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = - 0.2, 1$