Solve for x 16(1+3x)^2+11=36

$16 {(1 + 3 x)}^{2} + 11 = 36$

Step 1

Subtract $11$ from both sides of the equation.

$16 {(1 + 3 x)}^{2} = 36 - 11$

Step 2

Subtract $11$ from $36$ .

$16 {(1 + 3 x)}^{2} = 25$

Step 3

Divide each term in $16 {(1 + 3 x)}^{2} = 25$ by $16$ and simplify.

Divide each term in $16 {(1 + 3 x)}^{2} = 25$ by $16$ .

$\frac{16 {(1 + 3 x)}^{2}}{16} = \frac{25}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 {(1 + 3 x)}^{2}}{16} = \frac{25}{16}$

Divide ${(1 + 3 x)}^{2}$ by $1$ .

${(1 + 3 x)}^{2} = \frac{25}{16}$

Step 4

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$1 + 3 x = \pm \sqrt{\frac{25}{16}}$

Step 5

Simplify $\pm \sqrt{\frac{25}{16}}$ .

Rewrite $\sqrt{\frac{25}{16}}$ as $\frac{\sqrt{25}}{\sqrt{16}}$ .

$1 + 3 x = \pm \frac{\sqrt{25}}{\sqrt{16}}$

Simplify the numerator.

Rewrite $25$ as $5^{2}$ .

$1 + 3 x = \pm \frac{\sqrt{5^{2}}}{\sqrt{16}}$

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

$1 + 3 x = \pm \frac{5}{\sqrt{16}}$

Simplify the denominator.

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

$1 + 3 x = \pm \frac{5}{\sqrt{4^{2}}}$

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

$1 + 3 x = \pm \frac{5}{4}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$1 + 3 x = \frac{5}{4}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$3 x = \frac{5}{4} - 1$

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

$3 x = \frac{5}{4} - 1 \cdot \frac{4}{4}$

Combine $- 1$ and $\frac{4}{4}$ .

$3 x = \frac{5}{4} + \frac{- 1 \cdot 4}{4}$

Combine the numerators over the common denominator.

$3 x = \frac{5 - 1 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 1$ by $4$ .

$3 x = \frac{5 - 4}{4}$

Subtract $4$ from $5$ .

$3 x = \frac{1}{4}$

Divide each term in $3 x = \frac{1}{4}$ by $3$ and simplify.

Divide each term in $3 x = \frac{1}{4}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{1}{4}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{1}{4}}{3}$

Divide $x$ by $1$ .

$x = \frac{\frac{1}{4}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{4} \cdot \frac{1}{3}$

Multiply $\frac{1}{4} \cdot \frac{1}{3}$ .

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

$x = \frac{1}{4 \cdot 3}$

Multiply $4$ by $3$ .

$x = \frac{1}{12}$

Next, use the negative value of the $\pm$ to find the second solution.

$1 + 3 x = - \frac{5}{4}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$3 x = - \frac{5}{4} - 1$

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

$3 x = - \frac{5}{4} - 1 \cdot \frac{4}{4}$

Combine $- 1$ and $\frac{4}{4}$ .

$3 x = - \frac{5}{4} + \frac{- 1 \cdot 4}{4}$

Combine the numerators over the common denominator.

$3 x = \frac{- 5 - 1 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 1$ by $4$ .

$3 x = \frac{- 5 - 4}{4}$

Subtract $4$ from $- 5$ .

$3 x = \frac{- 9}{4}$

Move the negative in front of the fraction.

$3 x = - \frac{9}{4}$

Divide each term in $3 x = - \frac{9}{4}$ by $3$ and simplify.

Divide each term in $3 x = - \frac{9}{4}$ by $3$ .

$\frac{3 x}{3} = \frac{- \frac{9}{4}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- \frac{9}{4}}{3}$

Divide $x$ by $1$ .

$x = \frac{- \frac{9}{4}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{9}{4} \cdot \frac{1}{3}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$x = \frac{- 9}{4} \cdot \frac{1}{3}$

Factor $3$ out of $- 9$ .

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

Cancel the common factor.

$x = \frac{3 \cdot - 3}{4} \cdot \frac{1}{3}$

Rewrite the expression.

$x = \frac{- 3}{4}$

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1}{12}, - \frac{3}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{12}, - \frac{3}{4}$

Decimal Form:

$x = 0.08\overline{3}, - 0.75$

Solve for x 16(1+3x)^2+11=36

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