Basic Math Examples

$2 y^{2} + y + 19 = {(y - 3)}^{2}$

Step 1

Simplify ${(y - 3)}^{2}$ .

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Step 1.1

Rewrite.

$2 y^{2} + y + 19 = 0 + 0 + {(y - 3)}^{2}$

Step 1.2

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$2 y^{2} + y + 19 = (y - 3) (y - 3)$

Step 1.3

Expand $(y - 3) (y - 3)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$2 y^{2} + y + 19 = y (y - 3) - 3 (y - 3)$

Step 1.3.2

Apply the distributive property.

$2 y^{2} + y + 19 = y \cdot y + y \cdot - 3 - 3 (y - 3)$

Step 1.3.3

Apply the distributive property.

$2 y^{2} + y + 19 = y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $y$ by $y$ .

$2 y^{2} + y + 19 = y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3$

Step 1.4.1.2

Move $- 3$ to the left of $y$ .

$2 y^{2} + y + 19 = y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3$

Step 1.4.1.3

Multiply $- 3$ by $- 3$ .

$2 y^{2} + y + 19 = y^{2} - 3 y - 3 y + 9$

Step 1.4.2

Subtract $3 y$ from $- 3 y$ .

$2 y^{2} + y + 19 = y^{2} - 6 y + 9$

Step 2

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

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Step 2.1

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

$2 y^{2} + y + 19 - y^{2} = - 6 y + 9$

Step 2.2

Add $6 y$ to both sides of the equation.

$2 y^{2} + y + 19 - y^{2} + 6 y = 9$

Step 2.3

Subtract $y^{2}$ from $2 y^{2}$ .

$y^{2} + y + 19 + 6 y = 9$

Step 2.4

Add $y$ and $6 y$ .

$y^{2} + 7 y + 19 = 9$

Step 3

Subtract $9$ from both sides of the equation.

$y^{2} + 7 y + 19 - 9 = 0$

Step 4

Subtract $9$ from $19$ .

$y^{2} + 7 y + 10 = 0$

Step 5

Factor $y^{2} + 7 y + 10$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $7$ .

$2, 5$

Step 5.2

Write the factored form using these integers.

$(y + 2) (y + 5) = 0$

Step 6

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

$y + 2 = 0$

$y + 5 = 0$

Step 7

Set $y + 2$ equal to $0$ and solve for $y$ .

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Step 7.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$y = - 2$

Step 8

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

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Step 8.1

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

$y + 5 = 0$

Step 8.2

Subtract $5$ from both sides of the equation.

$y = - 5$

Step 9

The final solution is all the values that make $(y + 2) (y + 5) = 0$ true.

$y = - 2, - 5$