Basic Math Examples

$3 y^{2} = 11 y - 6$

Step 1

Subtract $11 y$ from both sides of the equation.

$3 y^{2} - 11 y = - 6$

Step 2

Add $6$ to both sides of the equation.

$3 y^{2} - 11 y + 6 = 0$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 3 \cdot 6 = 18$ and whose sum is $b = - 11$ .

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Step 3.1.1

Factor $- 11$ out of $- 11 y$ .

$3 y^{2} - 11 y + 6 = 0$

Step 3.1.2

Rewrite $- 11$ as $- 2$ plus $- 9$

$3 y^{2} + (- 2 - 9) y + 6 = 0$

Step 3.1.3

Apply the distributive property.

$3 y^{2} - 2 y - 9 y + 6 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(3 y^{2} - 2 y) - 9 y + 6 = 0$

Step 3.2.2

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

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

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 y - 2$ .

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

Step 4

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

$3 y - 2 = 0$

$y - 3 = 0$

Step 5

Set $3 y - 2$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $3 y - 2$ equal to $0$ .

$3 y - 2 = 0$

Step 5.2

Solve $3 y - 2 = 0$ for $y$ .

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Step 5.2.1

Add $2$ to both sides of the equation.

$3 y = 2$

Step 5.2.2

Divide each term in $3 y = 2$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 y = 2$ by $3$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 6

Set $y - 3$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Step 6.2

Add $3$ to both sides of the equation.

$y = 3$

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = 0.\overline{6}, 3$