Solve Using the Quadratic Formula 1/6x^2-1/3x=1

$\frac{1}{6} x^{2} - \frac{1}{3} x = 1$

Step 1

Move all terms to the left side of the equation and simplify.

Simplify the left side.

Simplify each term.

Combine $\frac{1}{6}$ and $x^{2}$ .

$\frac{x^{2}}{6} - \frac{1}{3} x = 1$

Combine $x$ and $\frac{1}{3}$ .

$\frac{x^{2}}{6} - \frac{x}{3} = 1$

Subtract $1$ from both sides of the equation.

$\frac{x^{2}}{6} - \frac{x}{3} - 1 = 0$

Step 2

Multiply through by the least common denominator $6$ , then simplify.

Apply the distributive property.

$6 (\frac{x^{2}}{6}) + 6 (- \frac{x}{3}) + 6 \cdot - 1 = 0$

Simplify.

Cancel the common factor of $6$ .

Cancel the common factor.

$6 (\frac{x^{2}}{6}) + 6 (- \frac{x}{3}) + 6 \cdot - 1 = 0$

Rewrite the expression.

$x^{2} + 6 (- \frac{x}{3}) + 6 \cdot - 1 = 0$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$x^{2} + 6 (\frac{- x}{3}) + 6 \cdot - 1 = 0$

Factor $3$ out of $6$ .

$x^{2} + 3 (2) (\frac{- x}{3}) + 6 \cdot - 1 = 0$

Cancel the common factor.

$x^{2} + 3 \cdot (2 (\frac{- x}{3})) + 6 \cdot - 1 = 0$

Rewrite the expression.

$x^{2} + 2 (- x) + 6 \cdot - 1 = 0$

Multiply $- 1$ by $2$ .

$x^{2} - 2 x + 6 \cdot - 1 = 0$

Multiply $6$ by $- 1$ .

$x^{2} - 2 x - 6 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 6)}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 6$ .

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4$ by $- 6$ .

$x = \frac{2 \pm \sqrt{4 + 24}}{2 \cdot 1}$

Add $4$ and $24$ .

$x = \frac{2 \pm \sqrt{28}}{2 \cdot 1}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$x = \frac{2 \pm \sqrt{4 (7)}}{2 \cdot 1}$

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{7}}{2}$

Simplify $\frac{2 \pm 2 \sqrt{7}}{2}$ .

$x = 1 \pm \sqrt{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 1 \pm \sqrt{7}$

Decimal Form:

$x = 3.64575131 \dots, - 1.64575131 \dots$

Solve Using the Quadratic Formula 1/6x^2-1/3x=1

Related Questions

Support LCMGCF.com by disabling your adblocker.