Algebra Examples

$13^{2} = 144 + {y_{2}}^{2} - 12 y_{2} + 36$

Step 1

Rewrite the equation as $144 + {y_{2}}^{2} - 12 y_{2} + 36 = 13^{2}$ .

$144 + {y_{2}}^{2} - 12 y_{2} + 36 = 13^{2}$

Step 2

Add $144$ and $36$ .

${y_{2}}^{2} - 12 y_{2} + 180 = 13^{2}$

Step 3

Raise $13$ to the power of $2$ .

${y_{2}}^{2} - 12 y_{2} + 180 = 169$

Step 4

Subtract $169$ from both sides of the equation.

${y_{2}}^{2} - 12 y_{2} + 180 - 169 = 0$

Step 5

Subtract $169$ from $180$ .

${y_{2}}^{2} - 12 y_{2} + 11 = 0$

Step 6

Factor ${y_{2}}^{2} - 12 y_{2} + 11$ using the AC method.

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Step 6.1

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 $11$ and whose sum is $- 12$ .

$- 11, - 1$

Step 6.2

Write the factored form using these integers.

$(y_{2} - 11) (y_{2} - 1) = 0$

Step 7

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

$y_{2} - 11 = 0$

$y_{2} - 1 = 0$

Step 8

Set $y_{2} - 11$ equal to $0$ and solve for $y_{2}$ .

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Step 8.1

Set $y_{2} - 11$ equal to $0$ .

$y_{2} - 11 = 0$

Step 8.2

Add $11$ to both sides of the equation.

$y_{2} = 11$

Step 9

Set $y_{2} - 1$ equal to $0$ and solve for $y_{2}$ .

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Step 9.1

Set $y_{2} - 1$ equal to $0$ .

$y_{2} - 1 = 0$

Step 9.2

Add $1$ to both sides of the equation.

$y_{2} = 1$

Step 10

The final solution is all the values that make $(y_{2} - 11) (y_{2} - 1) = 0$ true.

$y_{2} = 11, 1$