Solve by Completing the Square 3x^2-16x-1=0

$3 x^{2} - 16 x - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$3 x^{2} - 16 x = 1$

Step 2

Divide each term in $3 x^{2} - 16 x = 1$ by $3$ and simplify.

Divide each term in $3 x^{2} - 16 x = 1$ by $3$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Move the negative in front of the fraction.

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{8}{3})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} - \frac{16 x}{3} + {(- \frac{8}{3})}^{2} = \frac{1}{3} + {(- \frac{8}{3})}^{2}$

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 $- \frac{8}{3}$ .

$x^{2} - \frac{16 x}{3} + {(- 1)}^{2} {(\frac{8}{3})}^{2} = \frac{1}{3} + {(- \frac{8}{3})}^{2}$

Apply the product rule to $\frac{8}{3}$ .

$x^{2} - \frac{16 x}{3} + {(- 1)}^{2} \frac{8^{2}}{3^{2}} = \frac{1}{3} + {(- \frac{8}{3})}^{2}$

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

$x^{2} - \frac{16 x}{3} + 1 \frac{8^{2}}{3^{2}} = \frac{1}{3} + {(- \frac{8}{3})}^{2}$

Multiply $\frac{8^{2}}{3^{2}}$ by $1$ .

$x^{2} - \frac{16 x}{3} + \frac{8^{2}}{3^{2}} = \frac{1}{3} + {(- \frac{8}{3})}^{2}$

Raise $8$ to the power of $2$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{3^{2}} = \frac{1}{3} + {(- \frac{8}{3})}^{2}$

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

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{1}{3} + {(- \frac{8}{3})}^{2}$

Simplify the right side.

Simplify $\frac{1}{3} + {(- \frac{8}{3})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{8}{3}$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{1}{3} + {(- 1)}^{2} {(\frac{8}{3})}^{2}$

Apply the product rule to $\frac{8}{3}$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{1}{3} + {(- 1)}^{2} \frac{8^{2}}{3^{2}}$

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

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{1}{3} + 1 \frac{8^{2}}{3^{2}}$

Multiply $\frac{8^{2}}{3^{2}}$ by $1$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{1}{3} + \frac{8^{2}}{3^{2}}$

Raise $8$ to the power of $2$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{1}{3} + \frac{64}{3^{2}}$

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

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{1}{3} + \frac{64}{9}$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{1}{3} \cdot \frac{3}{3} + \frac{64}{9}$

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3}$ by $\frac{3}{3}$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{3}{3 \cdot 3} + \frac{64}{9}$

Multiply $3$ by $3$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{3}{9} + \frac{64}{9}$

Combine the numerators over the common denominator.

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{3 + 64}{9}$

Add $3$ and $64$ .

$x^{2} - \frac{16 x}{3} + \frac{64}{9} = \frac{67}{9}$

Step 6

Factor the perfect trinomial square into ${(x - \frac{8}{3})}^{2}$ .

${(x - \frac{8}{3})}^{2} = \frac{67}{9}$

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.

$x - \frac{8}{3} = \pm \sqrt{\frac{67}{9}}$

Simplify $\pm \sqrt{\frac{67}{9}}$ .

Rewrite $\sqrt{\frac{67}{9}}$ as $\frac{\sqrt{67}}{\sqrt{9}}$ .

$x - \frac{8}{3} = \pm \frac{\sqrt{67}}{\sqrt{9}}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$x - \frac{8}{3} = \pm \frac{\sqrt{67}}{\sqrt{3^{2}}}$

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

$x - \frac{8}{3} = \pm \frac{\sqrt{67}}{3}$

Add $\frac{8}{3}$ to both sides of the equation.

$x = \pm \frac{\sqrt{67}}{3} + \frac{8}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{67}}{3} + \frac{8}{3}$

Decimal Form:

$x = 5.39511759 \dots, - 0.06178425 \dots$

Solve by Completing the Square 3x^2-16x-1=0

Related Questions

Support LCMGCF.com by disabling your adblocker.