Solve by Completing the Square 5x^2-3x=25

$5 x^{2} - 3 x = 25$

Step 1

Divide each term in $5 x^{2} - 3 x = 25$ by $5$ and simplify.

Divide each term in $5 x^{2} - 3 x = 25$ by $5$ .

$\frac{5 x^{2}}{5} + \frac{- 3 x}{5} = \frac{25}{5}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x^{2}}{5} + \frac{- 3 x}{5} = \frac{25}{5}$

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

$x^{2} + \frac{- 3 x}{5} = \frac{25}{5}$

Move the negative in front of the fraction.

$x^{2} - \frac{3 x}{5} = \frac{25}{5}$

Simplify the right side.

Divide $25$ by $5$ .

$x^{2} - \frac{3 x}{5} = 5$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{3}{10})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{3 x}{5} + {(- \frac{3}{10})}^{2} = 5 + {(- \frac{3}{10})}^{2}$

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 $- \frac{3}{10}$ .

$x^{2} - \frac{3 x}{5} + {(- 1)}^{2} {(\frac{3}{10})}^{2} = 5 + {(- \frac{3}{10})}^{2}$

Apply the product rule to $\frac{3}{10}$ .

$x^{2} - \frac{3 x}{5} + {(- 1)}^{2} \frac{3^{2}}{10^{2}} = 5 + {(- \frac{3}{10})}^{2}$

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

$x^{2} - \frac{3 x}{5} + 1 \frac{3^{2}}{10^{2}} = 5 + {(- \frac{3}{10})}^{2}$

Multiply $\frac{3^{2}}{10^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{5} + \frac{3^{2}}{10^{2}} = 5 + {(- \frac{3}{10})}^{2}$

Raise $3$ to the power of $2$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{10^{2}} = 5 + {(- \frac{3}{10})}^{2}$

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

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = 5 + {(- \frac{3}{10})}^{2}$

Simplify the right side.

Simplify $5 + {(- \frac{3}{10})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{3}{10}$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = 5 + {(- 1)}^{2} {(\frac{3}{10})}^{2}$

Apply the product rule to $\frac{3}{10}$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = 5 + {(- 1)}^{2} \frac{3^{2}}{10^{2}}$

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

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = 5 + 1 \frac{3^{2}}{10^{2}}$

Multiply $\frac{3^{2}}{10^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = 5 + \frac{3^{2}}{10^{2}}$

Raise $3$ to the power of $2$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = 5 + \frac{9}{10^{2}}$

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

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = 5 + \frac{9}{100}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{100}{100}$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = 5 \cdot \frac{100}{100} + \frac{9}{100}$

Combine $5$ and $\frac{100}{100}$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = \frac{5 \cdot 100}{100} + \frac{9}{100}$

Combine the numerators over the common denominator.

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = \frac{5 \cdot 100 + 9}{100}$

Simplify the numerator.

Multiply $5$ by $100$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = \frac{500 + 9}{100}$

Add $500$ and $9$ .

$x^{2} - \frac{3 x}{5} + \frac{9}{100} = \frac{509}{100}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{3}{10})}^{2}$ .

${(x - \frac{3}{10})}^{2} = \frac{509}{100}$

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.

$x - \frac{3}{10} = \pm \sqrt{\frac{509}{100}}$

Simplify $\pm \sqrt{\frac{509}{100}}$ .

Rewrite $\sqrt{\frac{509}{100}}$ as $\frac{\sqrt{509}}{\sqrt{100}}$ .

$x - \frac{3}{10} = \pm \frac{\sqrt{509}}{\sqrt{100}}$

Simplify the denominator.

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

$x - \frac{3}{10} = \pm \frac{\sqrt{509}}{\sqrt{10^{2}}}$

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

$x - \frac{3}{10} = \pm \frac{\sqrt{509}}{10}$

Add $\frac{3}{10}$ to both sides of the equation.

$x = \pm \frac{\sqrt{509}}{10} + \frac{3}{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{509}}{10} + \frac{3}{10}$

Decimal Form:

$x = 2.55610283 \dots, - 1.95610283 \dots$

Solve by Completing the Square 5x^2-3x=25

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