Basic Math Examples

${(v - 5)}^{2} = 2 v^{2} - 16 v + 33$

Step 1

Since $v$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 v^{2} - 16 v + 33 = {(v - 5)}^{2}$

Step 2

Simplify ${(v - 5)}^{2}$ .

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Step 2.1

Rewrite.

$2 v^{2} - 16 v + 33 = 0 + 0 + {(v - 5)}^{2}$

Step 2.2

Rewrite ${(v - 5)}^{2}$ as $(v - 5) (v - 5)$ .

$2 v^{2} - 16 v + 33 = (v - 5) (v - 5)$

Step 2.3

Expand $(v - 5) (v - 5)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$2 v^{2} - 16 v + 33 = v (v - 5) - 5 (v - 5)$

Step 2.3.2

Apply the distributive property.

$2 v^{2} - 16 v + 33 = v \cdot v + v \cdot - 5 - 5 (v - 5)$

Step 2.3.3

Apply the distributive property.

$2 v^{2} - 16 v + 33 = v \cdot v + v \cdot - 5 - 5 v - 5 \cdot - 5$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $v$ by $v$ .

$2 v^{2} - 16 v + 33 = v^{2} + v \cdot - 5 - 5 v - 5 \cdot - 5$

Step 2.4.1.2

Move $- 5$ to the left of $v$ .

$2 v^{2} - 16 v + 33 = v^{2} - 5 \cdot v - 5 v - 5 \cdot - 5$

Step 2.4.1.3

Multiply $- 5$ by $- 5$ .

$2 v^{2} - 16 v + 33 = v^{2} - 5 v - 5 v + 25$

Step 2.4.2

Subtract $5 v$ from $- 5 v$ .

$2 v^{2} - 16 v + 33 = v^{2} - 10 v + 25$

Step 3

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

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Step 3.1

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

$2 v^{2} - 16 v + 33 - v^{2} = - 10 v + 25$

Step 3.2

Add $10 v$ to both sides of the equation.

$2 v^{2} - 16 v + 33 - v^{2} + 10 v = 25$

Step 3.3

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

$v^{2} - 16 v + 33 + 10 v = 25$

Step 3.4

Add $- 16 v$ and $10 v$ .

$v^{2} - 6 v + 33 = 25$

Step 4

Subtract $25$ from both sides of the equation.

$v^{2} - 6 v + 33 - 25 = 0$

Step 5

Subtract $25$ from $33$ .

$v^{2} - 6 v + 8 = 0$

Step 6

Factor $v^{2} - 6 v + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 6.2

Write the factored form using these integers.

$(v - 4) (v - 2) = 0$

Step 7

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

$v - 4 = 0$

$v - 2 = 0$

Step 8

Set $v - 4$ equal to $0$ and solve for $v$ .

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Step 8.1

Set $v - 4$ equal to $0$ .

$v - 4 = 0$

Step 8.2

Add $4$ to both sides of the equation.

$v = 4$

Step 9

Set $v - 2$ equal to $0$ and solve for $v$ .

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Step 9.1

Set $v - 2$ equal to $0$ .

$v - 2 = 0$

Step 9.2

Add $2$ to both sides of the equation.

$v = 2$

Step 10

The final solution is all the values that make $(v - 4) (v - 2) = 0$ true.

$v = 4, 2$