Solve by Completing the Square 6x=4x^2-1

$6 x = 4 x^{2} - 1$

Step 1

Subtract $4 x^{2}$ from both sides of the equation.

$6 x - 4 x^{2} = - 1$

Step 2

Divide each term in $6 x - 4 x^{2} = - 1$ by $- 4$ and simplify.

Divide each term in $6 x - 4 x^{2} = - 1$ by $- 4$ .

$\frac{6 x}{- 4} + \frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $6$ and $- 4$ .

Factor $2$ out of $6 x$ .

$\frac{2 (3 x)}{- 4} + \frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Cancel the common factors.

Factor $2$ out of $- 4$ .

$\frac{2 (3 x)}{2 (- 2)} + \frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Cancel the common factor.

$\frac{2 (3 x)}{2 \cdot - 2} + \frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Rewrite the expression.

$\frac{3 x}{- 2} + \frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Move the negative in front of the fraction.

$- \frac{3 x}{2} + \frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Cancel the common factor of $- 4$ .

Cancel the common factor.

$- \frac{3 x}{2} + \frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

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

$- \frac{3 x}{2} + x^{2} = \frac{- 1}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

$- \frac{3 x}{2} + x^{2} = \frac{1}{4}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{3}{4})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} - \frac{3 x}{2} + {(- \frac{3}{4})}^{2} = \frac{1}{4} + {(- \frac{3}{4})}^{2}$

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 $- \frac{3}{4}$ .

$x^{2} - \frac{3 x}{2} + {(- 1)}^{2} {(\frac{3}{4})}^{2} = \frac{1}{4} + {(- \frac{3}{4})}^{2}$

Apply the product rule to $\frac{3}{4}$ .

$x^{2} - \frac{3 x}{2} + {(- 1)}^{2} \frac{3^{2}}{4^{2}} = \frac{1}{4} + {(- \frac{3}{4})}^{2}$

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

$x^{2} - \frac{3 x}{2} + 1 \frac{3^{2}}{4^{2}} = \frac{1}{4} + {(- \frac{3}{4})}^{2}$

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{2} + \frac{3^{2}}{4^{2}} = \frac{1}{4} + {(- \frac{3}{4})}^{2}$

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

$x^{2} - \frac{3 x}{2} + \frac{9}{4^{2}} = \frac{1}{4} + {(- \frac{3}{4})}^{2}$

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

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{1}{4} + {(- \frac{3}{4})}^{2}$

Simplify the right side.

Simplify $\frac{1}{4} + {(- \frac{3}{4})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{3}{4}$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{1}{4} + {(- 1)}^{2} {(\frac{3}{4})}^{2}$

Apply the product rule to $\frac{3}{4}$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{1}{4} + {(- 1)}^{2} \frac{3^{2}}{4^{2}}$

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

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{1}{4} + 1 \frac{3^{2}}{4^{2}}$

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{1}{4} + \frac{3^{2}}{4^{2}}$

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

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{1}{4} + \frac{9}{4^{2}}$

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

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{1}{4} + \frac{9}{16}$

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{1}{4} \cdot \frac{4}{4} + \frac{9}{16}$

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

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{4}{4 \cdot 4} + \frac{9}{16}$

Multiply $4$ by $4$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{4}{16} + \frac{9}{16}$

Combine the numerators over the common denominator.

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{4 + 9}{16}$

Add $4$ and $9$ .

$x^{2} - \frac{3 x}{2} + \frac{9}{16} = \frac{13}{16}$

Step 6

Factor the perfect trinomial square into ${(x - \frac{3}{4})}^{2}$ .

${(x - \frac{3}{4})}^{2} = \frac{13}{16}$

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.

$x - \frac{3}{4} = \pm \sqrt{\frac{13}{16}}$

Simplify $\pm \sqrt{\frac{13}{16}}$ .

Rewrite $\sqrt{\frac{13}{16}}$ as $\frac{\sqrt{13}}{\sqrt{16}}$ .

$x - \frac{3}{4} = \pm \frac{\sqrt{13}}{\sqrt{16}}$

Simplify the denominator.

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

$x - \frac{3}{4} = \pm \frac{\sqrt{13}}{\sqrt{4^{2}}}$

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

$x - \frac{3}{4} = \pm \frac{\sqrt{13}}{4}$

Add $\frac{3}{4}$ to both sides of the equation.

$x = \pm \frac{\sqrt{13}}{4} + \frac{3}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{13}}{4} + \frac{3}{4}$

Decimal Form:

$x = 1.65138781 \dots, - 0.15138781 \dots$

Solve by Completing the Square 6x=4x^2-1

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