Solve by Completing the Square 5x^2+8x+1=0

$5 x^{2} + 8 x + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$5 x^{2} + 8 x = - 1$

Step 2

Divide each term in $5 x^{2} + 8 x = - 1$ by $5$ and simplify.

Divide each term in $5 x^{2} + 8 x = - 1$ by $5$ .

$\frac{5 x^{2}}{5} + \frac{8 x}{5} = \frac{- 1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x^{2}}{5} + \frac{8 x}{5} = \frac{- 1}{5}$

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

$x^{2} + \frac{8 x}{5} = \frac{- 1}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} + \frac{8 x}{5} = - \frac{1}{5}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{4}{5})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{8 x}{5} + {(\frac{4}{5})}^{2} = - \frac{1}{5} + {(\frac{4}{5})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{4}{5}$ .

$x^{2} + \frac{8 x}{5} + \frac{4^{2}}{5^{2}} = - \frac{1}{5} + {(\frac{4}{5})}^{2}$

Raise $4$ to the power of $2$ .

$x^{2} + \frac{8 x}{5} + \frac{16}{5^{2}} = - \frac{1}{5} + {(\frac{4}{5})}^{2}$

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

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = - \frac{1}{5} + {(\frac{4}{5})}^{2}$

Simplify the right side.

Simplify $- \frac{1}{5} + {(\frac{4}{5})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{4}{5}$ .

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = - \frac{1}{5} + \frac{4^{2}}{5^{2}}$

Raise $4$ to the power of $2$ .

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = - \frac{1}{5} + \frac{16}{5^{2}}$

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

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = - \frac{1}{5} + \frac{16}{25}$

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = - \frac{1}{5} \cdot \frac{5}{5} + \frac{16}{25}$

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{5}$ by $\frac{5}{5}$ .

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = - \frac{5}{5 \cdot 5} + \frac{16}{25}$

Multiply $5$ by $5$ .

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = - \frac{5}{25} + \frac{16}{25}$

Combine the numerators over the common denominator.

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = \frac{- 5 + 16}{25}$

Add $- 5$ and $16$ .

$x^{2} + \frac{8 x}{5} + \frac{16}{25} = \frac{11}{25}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{4}{5})}^{2}$ .

${(x + \frac{4}{5})}^{2} = \frac{11}{25}$

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.

$x + \frac{4}{5} = \pm \sqrt{\frac{11}{25}}$

Simplify $\pm \sqrt{\frac{11}{25}}$ .

Rewrite $\sqrt{\frac{11}{25}}$ as $\frac{\sqrt{11}}{\sqrt{25}}$ .

$x + \frac{4}{5} = \pm \frac{\sqrt{11}}{\sqrt{25}}$

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

$x + \frac{4}{5} = \pm \frac{\sqrt{11}}{\sqrt{5^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$x + \frac{4}{5} = \pm \frac{\sqrt{11}}{5}$

Subtract $\frac{4}{5}$ from both sides of the equation.

$x = \pm \frac{\sqrt{11}}{5} - \frac{4}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{11}}{5} - \frac{4}{5}$

Decimal Form:

$x = - 0.13667504 \dots, - 1.46332495 \dots$

Solve by Completing the Square 5x^2+8x+1=0

Related Questions

Support LCMGCF.com by disabling your adblocker.