Solve by Completing the Square 4x^2+5=10x

$4 x^{2} + 5 = 10 x$

Step 1

Simplify the equation into a proper form to complete the square.

Subtract $5$ from both sides of the equation.

$4 x^{2} = 10 x - 5$

Subtract $10 x$ from both sides of the equation.

$4 x^{2} - 10 x = - 5$

Step 2

Divide each term in $4 x^{2} - 10 x = - 5$ by $4$ and simplify.

Divide each term in $4 x^{2} - 10 x = - 5$ by $4$ .

$\frac{4 x^{2}}{4} + \frac{- 10 x}{4} = \frac{- 5}{4}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} + \frac{- 10 x}{4} = \frac{- 5}{4}$

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

$x^{2} + \frac{- 10 x}{4} = \frac{- 5}{4}$

Cancel the common factor of $- 10$ and $4$ .

Factor $2$ out of $- 10 x$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Simplify the right side.

Move the negative in front of the fraction.

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

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 $b$ .

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

Step 4

Add the term to each side of the equation.

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

Step 5

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 $- \frac{5}{4}$ .

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

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

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

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

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

Multiply $\frac{5^{2}}{4^{2}}$ by $1$ .

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

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

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

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

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

Simplify the right side.

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

Simplify each term.

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

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

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

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

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

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

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

Multiply $\frac{5^{2}}{4^{2}}$ by $1$ .

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

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

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

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

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

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

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

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{4}$ by $\frac{4}{4}$ .

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

Multiply $4$ by $4$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 5$ by $4$ .

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = \frac{- 20 + 25}{16}$

Add $- 20$ and $25$ .

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

Step 6

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

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

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.

$x - \frac{5}{4} = \pm \sqrt{\frac{5}{16}}$

Simplify $\pm \sqrt{\frac{5}{16}}$ .

Rewrite $\sqrt{\frac{5}{16}}$ as $\frac{\sqrt{5}}{\sqrt{16}}$ .

$x - \frac{5}{4} = \pm \frac{\sqrt{5}}{\sqrt{16}}$

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

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

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

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

Add $\frac{5}{4}$ to both sides of the equation.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.80901699 \dots, 0.69098300 \dots$

Solve by Completing the Square 4x^2+5=10x

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