Solve for X square root of 4X-11-2 square root of X=1

$\sqrt{4 X - 11} - 2 \sqrt{X} = 1$

Step 1

Add $2 \sqrt{X}$ to both sides of the equation.

$\sqrt{4 X - 11} = 1 + 2 \sqrt{X}$

Step 2

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

${\sqrt{4 X - 11}}^{2} = {(1 + 2 \sqrt{X})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 X - 11}$ as ${(4 X - 11)}^{\frac{1}{2}}$ .

${({(4 X - 11)}^{\frac{1}{2}})}^{2} = {(1 + 2 \sqrt{X})}^{2}$

Simplify the left side.

Simplify ${({(4 X - 11)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 X - 11)}^{\frac{1}{2}})}^{2}$ .

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

${(4 X - 11)}^{\frac{1}{2} \cdot 2} = {(1 + 2 \sqrt{X})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 X - 11)}^{\frac{1}{2} \cdot 2} = {(1 + 2 \sqrt{X})}^{2}$

Rewrite the expression.

${(4 X - 11)}^{1} = {(1 + 2 \sqrt{X})}^{2}$

Simplify.

$4 X - 11 = {(1 + 2 \sqrt{X})}^{2}$

Simplify the right side.

Simplify ${(1 + 2 \sqrt{X})}^{2}$ .

Rewrite ${(1 + 2 \sqrt{X})}^{2}$ as $(1 + 2 \sqrt{X}) (1 + 2 \sqrt{X})$ .

$4 X - 11 = (1 + 2 \sqrt{X}) (1 + 2 \sqrt{X})$

Expand $(1 + 2 \sqrt{X}) (1 + 2 \sqrt{X})$ using the FOIL Method.

Apply the distributive property.

$4 X - 11 = 1 (1 + 2 \sqrt{X}) + 2 \sqrt{X} (1 + 2 \sqrt{X})$

Apply the distributive property.

$4 X - 11 = 1 \cdot 1 + 1 (2 \sqrt{X}) + 2 \sqrt{X} (1 + 2 \sqrt{X})$

Apply the distributive property.

$4 X - 11 = 1 \cdot 1 + 1 (2 \sqrt{X}) + 2 \sqrt{X} \cdot 1 + 2 \sqrt{X} (2 \sqrt{X})$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$4 X - 11 = 1 + 1 (2 \sqrt{X}) + 2 \sqrt{X} \cdot 1 + 2 \sqrt{X} (2 \sqrt{X})$

Multiply $2 \sqrt{X}$ by $1$ .

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} \cdot 1 + 2 \sqrt{X} (2 \sqrt{X})$

Multiply $2$ by $1$ .

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 2 \sqrt{X} (2 \sqrt{X})$

Multiply $2 \sqrt{X} (2 \sqrt{X})$ .

Multiply $2$ by $2$ .

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 \sqrt{X} \sqrt{X}$

Raise $\sqrt{X}$ to the power of $1$ .

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 ({\sqrt{X}}^{1} \sqrt{X})$

Raise $\sqrt{X}$ to the power of $1$ .

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 ({\sqrt{X}}^{1} {\sqrt{X}}^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 {\sqrt{X}}^{1 + 1}$

Add $1$ and $1$ .

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 {\sqrt{X}}^{2}$

Rewrite ${\sqrt{X}}^{2}$ as $X$ .

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

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 {(X^{\frac{1}{2}})}^{2}$

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

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 X^{\frac{1}{2} \cdot 2}$

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

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 X^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 X^{\frac{2}{2}}$

Rewrite the expression.

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 X^{1}$

Simplify.

$4 X - 11 = 1 + 2 \sqrt{X} + 2 \sqrt{X} + 4 X$

Add $2 \sqrt{X}$ and $2 \sqrt{X}$ .

$4 X - 11 = 1 + 4 \sqrt{X} + 4 X$

Step 4

Solve for $4 \sqrt{X}$ .

Rewrite the equation as $1 + 4 \sqrt{X} + 4 X = 4 X - 11$ .

$1 + 4 \sqrt{X} + 4 X = 4 X - 11$

Move all terms not containing $4 \sqrt{X}$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$4 \sqrt{X} + 4 X = 4 X - 11 - 1$

Subtract $4 X$ from both sides of the equation.

$4 \sqrt{X} = 4 X - 11 - 1 - 4 X$

Combine the opposite terms in $4 X - 11 - 1 - 4 X$ .

Subtract $4 X$ from $4 X$ .

$4 \sqrt{X} = 0 - 11 - 1$

Subtract $11$ from $0$ .

$4 \sqrt{X} = - 11 - 1$

Subtract $1$ from $- 11$ .

$4 \sqrt{X} = - 12$

Step 5

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

${(4 \sqrt{X})}^{2} = {(- 12)}^{2}$

Step 6

Simplify each side of the equation.

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

${(4 X^{\frac{1}{2}})}^{2} = {(- 12)}^{2}$

Simplify the left side.

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

Apply the product rule to $4 X^{\frac{1}{2}}$ .

$4^{2} {(X^{\frac{1}{2}})}^{2} = {(- 12)}^{2}$

Raise $4$ to the power of $2$ .

$16 {(X^{\frac{1}{2}})}^{2} = {(- 12)}^{2}$

Multiply the exponents in ${(X^{\frac{1}{2}})}^{2}$ .

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

$16 X^{\frac{1}{2} \cdot 2} = {(- 12)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 X^{\frac{1}{2} \cdot 2} = {(- 12)}^{2}$

Rewrite the expression.

$16 X^{1} = {(- 12)}^{2}$

Simplify.

$16 X = {(- 12)}^{2}$

Simplify the right side.

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

$16 X = 144$

Step 7

Divide each term in $16 X = 144$ by $16$ and simplify.

Divide each term in $16 X = 144$ by $16$ .

$\frac{16 X}{16} = \frac{144}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 X}{16} = \frac{144}{16}$

Divide $X$ by $1$ .

$X = \frac{144}{16}$

Simplify the right side.

Divide $144$ by $16$ .

$X = 9$

Step 8

Exclude the solutions that do not make $\sqrt{4 X - 11} - 2 \sqrt{X} = 1$ true.

$No solution$

Solve for X square root of 4X-11-2 square root of X=1

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