Solve for x square root of 4x+1- square root of 5-2x=0

$\sqrt{4 x + 1} - \sqrt{5 - 2 x} = 0$

Step 1

Add $\sqrt{5 - 2 x}$ to both sides of the equation.

$\sqrt{4 x + 1} = \sqrt{5 - 2 x}$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 1}$ as ${(4 x + 1)}^{\frac{1}{2}}$ .

${({(4 x + 1)}^{\frac{1}{2}})}^{2} = {\sqrt{5 - 2 x}}^{2}$

Simplify the left side.

Simplify ${({(4 x + 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 x + 1)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(4 x + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{5 - 2 x}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{5 - 2 x}}^{2}$

Rewrite the expression.

${(4 x + 1)}^{1} = {\sqrt{5 - 2 x}}^{2}$

Simplify.

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

Simplify the right side.

Rewrite ${\sqrt{5 - 2 x}}^{2}$ as $5 - 2 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 - 2 x}$ as ${(5 - 2 x)}^{\frac{1}{2}}$ .

$4 x + 1 = {({(5 - 2 x)}^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 x + 1 = {(5 - 2 x)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$4 x + 1 = {(5 - 2 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x + 1 = {(5 - 2 x)}^{\frac{2}{2}}$

Rewrite the expression.

$4 x + 1 = {(5 - 2 x)}^{1}$

Simplify.

$4 x + 1 = 5 - 2 x$

Step 4

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Add $2 x$ to both sides of the equation.

$4 x + 1 + 2 x = 5$

Add $4 x$ and $2 x$ .

$6 x + 1 = 5$

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

Subtract $1$ from both sides of the equation.

$6 x = 5 - 1$

Subtract $1$ from $5$ .

$6 x = 4$

Divide each term in $6 x = 4$ by $6$ and simplify.

Divide each term in $6 x = 4$ by $6$ .

$\frac{6 x}{6} = \frac{4}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{4}{6}$

Divide $x$ by $1$ .

$x = \frac{4}{6}$

Simplify the right side.

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = \frac{2 \cdot 2}{2 \cdot 3}$

Cancel the common factor.

$x = \frac{2 \cdot 2}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{3}$

Decimal Form:

$x = 0.\overline{6}$

Solve for x square root of 4x+1- square root of 5-2x=0

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