Solve by Completing the Square 5x^2-7x=1

$5 x^{2} - 7 x = 1$

Step 1

Divide each term in $5 x^{2} - 7 x = 1$ by $5$ and simplify.

Divide each term in $5 x^{2} - 7 x = 1$ by $5$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $5$ .

Cancel the common factor.

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

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

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

Move the negative in front of the fraction.

$x^{2} - \frac{7 x}{5} = \frac{1}{5}$

Step 2

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

${(\frac{b}{2})}^{2} = {(- \frac{7}{10})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{7 x}{5} + {(- \frac{7}{10})}^{2} = \frac{1}{5} + {(- \frac{7}{10})}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7}{10}$ .

$x^{2} - \frac{7 x}{5} + {(- 1)}^{2} {(\frac{7}{10})}^{2} = \frac{1}{5} + {(- \frac{7}{10})}^{2}$

Apply the product rule to $\frac{7}{10}$ .

$x^{2} - \frac{7 x}{5} + {(- 1)}^{2} \frac{7^{2}}{10^{2}} = \frac{1}{5} + {(- \frac{7}{10})}^{2}$

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

$x^{2} - \frac{7 x}{5} + 1 \frac{7^{2}}{10^{2}} = \frac{1}{5} + {(- \frac{7}{10})}^{2}$

Multiply $\frac{7^{2}}{10^{2}}$ by $1$ .

$x^{2} - \frac{7 x}{5} + \frac{7^{2}}{10^{2}} = \frac{1}{5} + {(- \frac{7}{10})}^{2}$

Raise $7$ to the power of $2$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{10^{2}} = \frac{1}{5} + {(- \frac{7}{10})}^{2}$

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

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{1}{5} + {(- \frac{7}{10})}^{2}$

Simplify the right side.

Simplify $\frac{1}{5} + {(- \frac{7}{10})}^{2}$ .

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7}{10}$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{1}{5} + {(- 1)}^{2} {(\frac{7}{10})}^{2}$

Apply the product rule to $\frac{7}{10}$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{1}{5} + {(- 1)}^{2} \frac{7^{2}}{10^{2}}$

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

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{1}{5} + 1 \frac{7^{2}}{10^{2}}$

Multiply $\frac{7^{2}}{10^{2}}$ by $1$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{1}{5} + \frac{7^{2}}{10^{2}}$

Raise $7$ to the power of $2$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{1}{5} + \frac{49}{10^{2}}$

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

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{1}{5} + \frac{49}{100}$

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{1}{5} \cdot \frac{20}{20} + \frac{49}{100}$

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

Multiply $\frac{1}{5}$ by $\frac{20}{20}$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{20}{5 \cdot 20} + \frac{49}{100}$

Multiply $5$ by $20$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{20}{100} + \frac{49}{100}$

Combine the numerators over the common denominator.

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{20 + 49}{100}$

Add $20$ and $49$ .

$x^{2} - \frac{7 x}{5} + \frac{49}{100} = \frac{69}{100}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{7}{10})}^{2}$ .

${(x - \frac{7}{10})}^{2} = \frac{69}{100}$

Step 6

Solve the equation for $x$ .

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

$x - \frac{7}{10} = \pm \sqrt{\frac{69}{100}}$

Simplify $\pm \sqrt{\frac{69}{100}}$ .

Rewrite $\sqrt{\frac{69}{100}}$ as $\frac{\sqrt{69}}{\sqrt{100}}$ .

$x - \frac{7}{10} = \pm \frac{\sqrt{69}}{\sqrt{100}}$

Simplify the denominator.

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

$x - \frac{7}{10} = \pm \frac{\sqrt{69}}{\sqrt{10^{2}}}$

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

$x - \frac{7}{10} = \pm \frac{\sqrt{69}}{10}$

Add $\frac{7}{10}$ to both sides of the equation.

$x = \pm \frac{\sqrt{69}}{10} + \frac{7}{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{69}}{10} + \frac{7}{10}$

Decimal Form:

$x = 1.53066238 \dots, - 0.13066238 \dots$

Solve by Completing the Square 5x^2-7x=1

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