Solve Using the Square Root Property 36x^2+12x+1=18

$36 x^{2} + 12 x + 1 = 18$

Step 1

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

Subtract $18$ from both sides of the equation.

$36 x^{2} + 12 x + 1 - 18 = 0$

Subtract $18$ from $1$ .

$36 x^{2} + 12 x - 17 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 36$ , $b = 12$ , and $c = - 17$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (36 \cdot - 17)}}{2 \cdot 36}$

Step 4

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 36 \cdot - 17}}{2 \cdot 36}$

Multiply $- 4 \cdot 36 \cdot - 17$ .

Multiply $- 4$ by $36$ .

$x = \frac{- 12 \pm \sqrt{144 - 144 \cdot - 17}}{2 \cdot 36}$

Multiply $- 144$ by $- 17$ .

$x = \frac{- 12 \pm \sqrt{144 + 2448}}{2 \cdot 36}$

Add $144$ and $2448$ .

$x = \frac{- 12 \pm \sqrt{2592}}{2 \cdot 36}$

Rewrite $2592$ as $36^{2} \cdot 2$ .

Factor $1296$ out of $2592$ .

$x = \frac{- 12 \pm \sqrt{1296 (2)}}{2 \cdot 36}$

Rewrite $1296$ as $36^{2}$ .

$x = \frac{- 12 \pm \sqrt{36^{2} \cdot 2}}{2 \cdot 36}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm 36 \sqrt{2}}{2 \cdot 36}$

Multiply $2$ by $36$ .

$x = \frac{- 12 \pm 36 \sqrt{2}}{72}$

Simplify $\frac{- 12 \pm 36 \sqrt{2}}{72}$ .

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

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 36 \cdot - 17}}{2 \cdot 36}$

Multiply $- 4 \cdot 36 \cdot - 17$ .

Multiply $- 4$ by $36$ .

$x = \frac{- 12 \pm \sqrt{144 - 144 \cdot - 17}}{2 \cdot 36}$

Multiply $- 144$ by $- 17$ .

$x = \frac{- 12 \pm \sqrt{144 + 2448}}{2 \cdot 36}$

Add $144$ and $2448$ .

$x = \frac{- 12 \pm \sqrt{2592}}{2 \cdot 36}$

Rewrite $2592$ as $36^{2} \cdot 2$ .

Factor $1296$ out of $2592$ .

$x = \frac{- 12 \pm \sqrt{1296 (2)}}{2 \cdot 36}$

Rewrite $1296$ as $36^{2}$ .

$x = \frac{- 12 \pm \sqrt{36^{2} \cdot 2}}{2 \cdot 36}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm 36 \sqrt{2}}{2 \cdot 36}$

Multiply $2$ by $36$ .

$x = \frac{- 12 \pm 36 \sqrt{2}}{72}$

Simplify $\frac{- 12 \pm 36 \sqrt{2}}{72}$ .

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

Change the $\pm$ to $+$ .

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

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + 3 \sqrt{2}}{6}$

Factor $- 1$ out of $3 \sqrt{2}$ .

$x = \frac{- 1 \cdot 1 - (- 3 \sqrt{2})}{6}$

Factor $- 1$ out of $- 1 (1) - (- 3 \sqrt{2})$ .

$x = \frac{- 1 (1 - 3 \sqrt{2})}{6}$

Move the negative in front of the fraction.

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

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 36 \cdot - 17}}{2 \cdot 36}$

Multiply $- 4 \cdot 36 \cdot - 17$ .

Multiply $- 4$ by $36$ .

$x = \frac{- 12 \pm \sqrt{144 - 144 \cdot - 17}}{2 \cdot 36}$

Multiply $- 144$ by $- 17$ .

$x = \frac{- 12 \pm \sqrt{144 + 2448}}{2 \cdot 36}$

Add $144$ and $2448$ .

$x = \frac{- 12 \pm \sqrt{2592}}{2 \cdot 36}$

Rewrite $2592$ as $36^{2} \cdot 2$ .

Factor $1296$ out of $2592$ .

$x = \frac{- 12 \pm \sqrt{1296 (2)}}{2 \cdot 36}$

Rewrite $1296$ as $36^{2}$ .

$x = \frac{- 12 \pm \sqrt{36^{2} \cdot 2}}{2 \cdot 36}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm 36 \sqrt{2}}{2 \cdot 36}$

Multiply $2$ by $36$ .

$x = \frac{- 12 \pm 36 \sqrt{2}}{72}$

Simplify $\frac{- 12 \pm 36 \sqrt{2}}{72}$ .

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

Change the $\pm$ to $-$ .

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

Rewrite $- 1$ as $- 1 (1)$ .

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

Factor $- 1$ out of $- 3 \sqrt{2}$ .

$x = \frac{- 1 \cdot 1 - (3 \sqrt{2})}{6}$

Factor $- 1$ out of $- 1 (1) - (3 \sqrt{2})$ .

$x = \frac{- 1 (1 + 3 \sqrt{2})}{6}$

Move the negative in front of the fraction.

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

Step 7

The final answer is the combination of both solutions.

$x = - \frac{1 - 3 \sqrt{2}}{6}, - \frac{1 + 3 \sqrt{2}}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - 3 \sqrt{2}}{6}, - \frac{1 + 3 \sqrt{2}}{6}$

Decimal Form:

$x = 0.54044011 \dots, - 0.87377344 \dots$

Solve Using the Square Root Property 36x^2+12x+1=18

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