Solve by Completing the Square 5x^2+20x+13=0

$5 x^{2} + 20 x + 13 = 0$

Step 1

Subtract $13$ from both sides of the equation.

$5 x^{2} + 20 x = - 13$

Step 2

Divide each term in $5 x^{2} + 20 x = - 13$ by $5$ and simplify.

Divide each term in $5 x^{2} + 20 x = - 13$ by $5$ .

$\frac{5 x^{2}}{5} + \frac{20 x}{5} = \frac{- 13}{5}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x^{2}}{5} + \frac{20 x}{5} = \frac{- 13}{5}$

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

$x^{2} + \frac{20 x}{5} = \frac{- 13}{5}$

Cancel the common factor of $20$ and $5$ .

Factor $5$ out of $20 x$ .

$x^{2} + \frac{5 (4 x)}{5} = \frac{- 13}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$x^{2} + \frac{5 (4 x)}{5 (1)} = \frac{- 13}{5}$

Cancel the common factor.

$x^{2} + \frac{5 (4 x)}{5 \cdot 1} = \frac{- 13}{5}$

Rewrite the expression.

$x^{2} + \frac{4 x}{1} = \frac{- 13}{5}$

Divide $4 x$ by $1$ .

$x^{2} + 4 x = \frac{- 13}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} + 4 x = - \frac{13}{5}$

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 $b$ .

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

Step 4

Add the term to each side of the equation.

$x^{2} + 4 x + {(2)}^{2} = - \frac{13}{5} + {(2)}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

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

$x^{2} + 4 x + 4 = - \frac{13}{5} + {(2)}^{2}$

Simplify the right side.

Simplify $- \frac{13}{5} + {(2)}^{2}$ .

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

$x^{2} + 4 x + 4 = - \frac{13}{5} + 4$

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{2} + 4 x + 4 = - \frac{13}{5} + 4 \cdot \frac{5}{5}$

Combine $4$ and $\frac{5}{5}$ .

$x^{2} + 4 x + 4 = - \frac{13}{5} + \frac{4 \cdot 5}{5}$

Combine the numerators over the common denominator.

$x^{2} + 4 x + 4 = \frac{- 13 + 4 \cdot 5}{5}$

Simplify the numerator.

Multiply $4$ by $5$ .

$x^{2} + 4 x + 4 = \frac{- 13 + 20}{5}$

Add $- 13$ and $20$ .

$x^{2} + 4 x + 4 = \frac{7}{5}$

Step 6

Factor the perfect trinomial square into ${(x + 2)}^{2}$ .

${(x + 2)}^{2} = \frac{7}{5}$

Step 7

Solve the equation for $x$ .

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

$x + 2 = \pm \sqrt{\frac{7}{5}}$

Simplify $\pm \sqrt{\frac{7}{5}}$ .

Rewrite $\sqrt{\frac{7}{5}}$ as $\frac{\sqrt{7}}{\sqrt{5}}$ .

$x + 2 = \pm \frac{\sqrt{7}}{\sqrt{5}}$

Multiply $\frac{\sqrt{7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x + 2 = \pm \frac{\sqrt{7}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

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

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

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

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{{\sqrt{5}}^{2}}$

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

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

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$x + 2 = \pm \frac{\sqrt{7} \sqrt{5}}{5}$

Simplify the numerator.

Combine using the product rule for radicals.

$x + 2 = \pm \frac{\sqrt{7 \cdot 5}}{5}$

Multiply $7$ by $5$ .

$x + 2 = \pm \frac{\sqrt{35}}{5}$

Subtract $2$ from both sides of the equation.

$x = \pm \frac{\sqrt{35}}{5} - 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{35}}{5} - 2$

Decimal Form:

$x = - 0.81678404 \dots, - 3.18321595 \dots$

Solve by Completing the Square 5x^2+20x+13=0

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