Solve by Completing the Square 5v^2-21=10v

$5 v^{2} - 21 = 10 v$

Step 1

Simplify the equation into a proper form to complete the square.

Add $21$ to both sides of the equation.

$5 v^{2} = 10 v + 21$

Subtract $10 v$ from both sides of the equation.

$5 v^{2} - 10 v = 21$

Step 2

Divide each term in $5 v^{2} - 10 v = 21$ by $5$ and simplify.

Divide each term in $5 v^{2} - 10 v = 21$ by $5$ .

$\frac{5 v^{2}}{5} + \frac{- 10 v}{5} = \frac{21}{5}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 v^{2}}{5} + \frac{- 10 v}{5} = \frac{21}{5}$

Divide $v^{2}$ by $1$ .

$v^{2} + \frac{- 10 v}{5} = \frac{21}{5}$

Cancel the common factor of $- 10$ and $5$ .

Factor $5$ out of $- 10 v$ .

$v^{2} + \frac{5 (- 2 v)}{5} = \frac{21}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$v^{2} + \frac{5 (- 2 v)}{5 (1)} = \frac{21}{5}$

Cancel the common factor.

$v^{2} + \frac{5 (- 2 v)}{5 \cdot 1} = \frac{21}{5}$

Rewrite the expression.

$v^{2} + \frac{- 2 v}{1} = \frac{21}{5}$

Divide $- 2 v$ by $1$ .

$v^{2} - 2 v = \frac{21}{5}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- 1)}^{2}$

Step 4

Add the term to each side of the equation.

$v^{2} - 2 v + {(- 1)}^{2} = \frac{21}{5} + {(- 1)}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Raise $- 1$ to the power of $2$ .

$v^{2} - 2 v + 1 = \frac{21}{5} + {(- 1)}^{2}$

Simplify the right side.

Simplify $\frac{21}{5} + {(- 1)}^{2}$ .

Raise $- 1$ to the power of $2$ .

$v^{2} - 2 v + 1 = \frac{21}{5} + 1$

Write $1$ as a fraction with a common denominator.

$v^{2} - 2 v + 1 = \frac{21}{5} + \frac{5}{5}$

Combine the numerators over the common denominator.

$v^{2} - 2 v + 1 = \frac{21 + 5}{5}$

Add $21$ and $5$ .

$v^{2} - 2 v + 1 = \frac{26}{5}$

Step 6

Factor the perfect trinomial square into ${(v - 1)}^{2}$ .

${(v - 1)}^{2} = \frac{26}{5}$

Step 7

Solve the equation for $v$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$v - 1 = \pm \sqrt{\frac{26}{5}}$

Simplify $\pm \sqrt{\frac{26}{5}}$ .

Rewrite $\sqrt{\frac{26}{5}}$ as $\frac{\sqrt{26}}{\sqrt{5}}$ .

$v - 1 = \pm \frac{\sqrt{26}}{\sqrt{5}}$

Multiply $\frac{\sqrt{26}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$v - 1 = \pm \frac{\sqrt{26}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{26}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$v - 1 = \pm \frac{\sqrt{26} \sqrt{5}}{5}$

Simplify the numerator.

Combine using the product rule for radicals.

$v - 1 = \pm \frac{\sqrt{26 \cdot 5}}{5}$

Multiply $26$ by $5$ .

$v - 1 = \pm \frac{\sqrt{130}}{5}$

Add $1$ to both sides of the equation.

$v = \pm \frac{\sqrt{130}}{5} + 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$v = \pm \frac{\sqrt{130}}{5} + 1$

Decimal Form:

$v = 3.28035085 \dots, - 1.28035085 \dots$

Solve by Completing the Square 5v^2-21=10v

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