Basic Math Examples

$2 v - 11 v + 16 = {(v - 2)}^{2}$

Step 1

Subtract $11 v$ from $2 v$ .

$- 9 v + 16 = {(v - 2)}^{2}$

Step 2

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

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Step 2.1

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

$- 9 v + 16 = (v - 2) (v - 2)$

Step 2.2

Expand $(v - 2) (v - 2)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$- 9 v + 16 = v (v - 2) - 2 (v - 2)$

Step 2.2.2

Apply the distributive property.

$- 9 v + 16 = v \cdot v + v \cdot - 2 - 2 (v - 2)$

Step 2.2.3

Apply the distributive property.

$- 9 v + 16 = v \cdot v + v \cdot - 2 - 2 v - 2 \cdot - 2$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $v$ by $v$ .

$- 9 v + 16 = v^{2} + v \cdot - 2 - 2 v - 2 \cdot - 2$

Step 2.3.1.2

Move $- 2$ to the left of $v$ .

$- 9 v + 16 = v^{2} - 2 \cdot v - 2 v - 2 \cdot - 2$

Step 2.3.1.3

Multiply $- 2$ by $- 2$ .

$- 9 v + 16 = v^{2} - 2 v - 2 v + 4$

Step 2.3.2

Subtract $2 v$ from $- 2 v$ .

$- 9 v + 16 = v^{2} - 4 v + 4$

Step 3

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

$v^{2} - 4 v + 4 = - 9 v + 16$

Step 4

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

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Step 4.1

Add $9 v$ to both sides of the equation.

$v^{2} - 4 v + 4 + 9 v = 16$

Step 4.2

Add $- 4 v$ and $9 v$ .

$v^{2} + 5 v + 4 = 16$

Step 5

Move all terms to the left side of the equation and simplify.

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Step 5.1

Subtract $16$ from both sides of the equation.

$v^{2} + 5 v + 4 - 16 = 0$

Step 5.2

Subtract $16$ from $4$ .

$v^{2} + 5 v - 12 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = 1$ , $b = 5$ , and $c = - 12$ into the quadratic formula and solve for $v$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $5$ to the power of $2$ .

$v = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 12$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$v = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 12}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 12$ .

$v = \frac{- 5 \pm \sqrt{25 + 48}}{2 \cdot 1}$

Step 8.1.3

Add $25$ and $48$ .

$v = \frac{- 5 \pm \sqrt{73}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$v = \frac{- 5 \pm \sqrt{73}}{2}$

Step 9

The final answer is the combination of both solutions.

$v = - \frac{5 - \sqrt{73}}{2}, - \frac{5 + \sqrt{73}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$v = - \frac{5 - \sqrt{73}}{2}, - \frac{5 + \sqrt{73}}{2}$

Decimal Form:

$v = 1.77200187 \dots, - 6.77200187 \dots$