Solve for x 16(4x-1)^2=25

$16 {(4 x - 1)}^{2} = 25$

Step 1

Divide each term in $16 {(4 x - 1)}^{2} = 25$ by $16$ and simplify.

Divide each term in $16 {(4 x - 1)}^{2} = 25$ by $16$ .

$\frac{16 {(4 x - 1)}^{2}}{16} = \frac{25}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 {(4 x - 1)}^{2}}{16} = \frac{25}{16}$

Divide ${(4 x - 1)}^{2}$ by $1$ .

${(4 x - 1)}^{2} = \frac{25}{16}$

Step 2

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

$4 x - 1 = \pm \sqrt{\frac{25}{16}}$

Step 3

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

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

$4 x - 1 = \pm \frac{\sqrt{25}}{\sqrt{16}}$

Simplify the numerator.

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

$4 x - 1 = \pm \frac{\sqrt{5^{2}}}{\sqrt{16}}$

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

$4 x - 1 = \pm \frac{5}{\sqrt{16}}$

Simplify the denominator.

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

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

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

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

Step 4

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.

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

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

Add $1$ to both sides of the equation.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $5$ and $4$ .

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

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{9}{4} \cdot \frac{1}{4}$ .

Multiply $\frac{9}{4}$ by $\frac{1}{4}$ .

$x = \frac{9}{4 \cdot 4}$

Multiply $4$ by $4$ .

$x = \frac{9}{16}$

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

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

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

Add $1$ to both sides of the equation.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $- 5$ and $4$ .

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

Move the negative in front of the fraction.

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

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $4$ by $4$ .

$x = - \frac{1}{16}$

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

$x = \frac{9}{16}, - \frac{1}{16}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{16}, - \frac{1}{16}$

Decimal Form:

$x = 0.5625, - 0.0625$

Solve for x 16(4x-1)^2=25

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