Solve by Completing the Square 5x^2+x-1/10=0

$5 x^{2} + x - \frac{1}{10} = 0$

Step 1

Add $\frac{1}{10}$ to both sides of the equation.

$5 x^{2} + x = \frac{1}{10}$

Step 2

Divide each term in $5 x^{2} + x = \frac{1}{10}$ by $5$ and simplify.

Divide each term in $5 x^{2} + x = \frac{1}{10}$ by $5$ .

$\frac{5 x^{2}}{5} + \frac{x}{5} = \frac{\frac{1}{10}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x^{2}}{5} + \frac{x}{5} = \frac{\frac{1}{10}}{5}$

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

$x^{2} + \frac{x}{5} = \frac{\frac{1}{10}}{5}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x^{2} + \frac{x}{5} = \frac{1}{10} \cdot \frac{1}{5}$

Multiply $\frac{1}{10} \cdot \frac{1}{5}$ .

Multiply $\frac{1}{10}$ by $\frac{1}{5}$ .

$x^{2} + \frac{x}{5} = \frac{1}{10 \cdot 5}$

Multiply $10$ by $5$ .

$x^{2} + \frac{x}{5} = \frac{1}{50}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{1}{10})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{x}{5} + {(\frac{1}{10})}^{2} = \frac{1}{50} + {(\frac{1}{10})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{1}{10}$ .

$x^{2} + \frac{x}{5} + \frac{1^{2}}{10^{2}} = \frac{1}{50} + {(\frac{1}{10})}^{2}$

One to any power is one.

$x^{2} + \frac{x}{5} + \frac{1}{10^{2}} = \frac{1}{50} + {(\frac{1}{10})}^{2}$

Raise $10$ to the power of $2$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{50} + {(\frac{1}{10})}^{2}$

Simplify the right side.

Simplify $\frac{1}{50} + {(\frac{1}{10})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{1}{10}$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{50} + \frac{1^{2}}{10^{2}}$

One to any power is one.

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{50} + \frac{1}{10^{2}}$

Raise $10$ to the power of $2$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{50} + \frac{1}{100}$

To write $\frac{1}{50}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{1}{50} \cdot \frac{2}{2} + \frac{1}{100}$

Write each expression with a common denominator of $100$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{50}$ by $\frac{2}{2}$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{2}{50 \cdot 2} + \frac{1}{100}$

Multiply $50$ by $2$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{2}{100} + \frac{1}{100}$

Combine the numerators over the common denominator.

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{2 + 1}{100}$

Add $2$ and $1$ .

$x^{2} + \frac{x}{5} + \frac{1}{100} = \frac{3}{100}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{1}{10})}^{2}$ .

${(x + \frac{1}{10})}^{2} = \frac{3}{100}$

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.

$x + \frac{1}{10} = \pm \sqrt{\frac{3}{100}}$

Simplify $\pm \sqrt{\frac{3}{100}}$ .

Rewrite $\sqrt{\frac{3}{100}}$ as $\frac{\sqrt{3}}{\sqrt{100}}$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{3}}{\sqrt{100}}$

Simplify the denominator.

Rewrite $100$ as $10^{2}$ .

$x + \frac{1}{10} = \pm \frac{\sqrt{3}}{\sqrt{10^{2}}}$

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

$x + \frac{1}{10} = \pm \frac{\sqrt{3}}{10}$

Subtract $\frac{1}{10}$ from both sides of the equation.

$x = \pm \frac{\sqrt{3}}{10} - \frac{1}{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{3}}{10} - \frac{1}{10}$

Decimal Form:

$x = 0.07320508 \dots, - 0.27320508 \dots$

Solve by Completing the Square 5x^2+x-1/10=0

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