Basic Math Examples

2 y^{2} - y = 6

$2 y^{2} - y = 6$

Step 1

Subtract

$6$ from both sides of the equation.

$2 y^{2} - y - 6 = 0$

Step 2

Factor by grouping.

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Step 2.1

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

$a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is

$b = - 1$ .

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Step 2.1.1

Factor

$- 1$ out of

$- y$ .

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

Step 2.1.2

Rewrite

$- 1$ as

$3$ plus

$- 4$

$2 y^{2} + (3 - 4) y - 6 = 0$

Step 2.1.3

Apply the distributive property.

$2 y^{2} + 3 y - 4 y - 6 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(2 y^{2} + 3 y) - 4 y - 6 = 0$

Step 2.2.2

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

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

Step 2.3

Factor the polynomial by factoring out the greatest common factor,

$2 y + 3$ .

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

Step 3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$2 y + 3 = 0$

$y - 2 = 0$

Step 4

Set

$2 y + 3$ equal to

$0$ and solve for

$y$ .

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Step 4.1

Set

$2 y + 3$ equal to

$0$ .

$2 y + 3 = 0$

Step 4.2

Solve

$2 y + 3 = 0$ for

$y$ .

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Step 4.2.1

Subtract

$3$ from both sides of the equation.

$2 y = - 3$

Step 4.2.2

Divide each term in

$2 y = - 3$ by

$2$ and simplify.

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Step 4.2.2.1

Divide each term in

$2 y = - 3$ by

$2$ .

$\frac{2 y}{2} = \frac{- 3}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of

$2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 3}{2}$

Step 4.2.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{- 3}{2}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$y = - \frac{3}{2}$

Step 5

Set

$y - 2$ equal to

$0$ and solve for

$y$ .

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Step 5.1

Set

$y - 2$ equal to

$0$ .

$y - 2 = 0$

Step 5.2

Add

$2$ to both sides of the equation.

$y = 2$

Step 6

The final solution is all the values that make

$(2 y + 3) (y - 2) = 0$ true.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = - 1.5, 2$

Mixed Number Form:

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