Solve for x Solve square root of 2x+21 = square root of 2x+3

Solve $\sqrt{2 x + 21} = \sqrt{2 x} + 3$

Step 1

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

${\sqrt{2 x + 21}}^{2} = {(\sqrt{2 x} + 3)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 21}$ as ${(2 x + 21)}^{\frac{1}{2}}$ .

${({(2 x + 21)}^{\frac{1}{2}})}^{2} = {(\sqrt{2 x} + 3)}^{2}$

Simplify the left side.

Simplify ${({(2 x + 21)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x + 21)}^{\frac{1}{2}})}^{2}$ .

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

${(2 x + 21)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 x} + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 21)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 x} + 3)}^{2}$

Rewrite the expression.

${(2 x + 21)}^{1} = {(\sqrt{2 x} + 3)}^{2}$

Simplify.

$2 x + 21 = {(\sqrt{2 x} + 3)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{2 x} + 3)}^{2}$ .

Rewrite ${(\sqrt{2 x} + 3)}^{2}$ as $(\sqrt{2 x} + 3) (\sqrt{2 x} + 3)$ .

$2 x + 21 = (\sqrt{2 x} + 3) (\sqrt{2 x} + 3)$

Expand $(\sqrt{2 x} + 3) (\sqrt{2 x} + 3)$ using the FOIL Method.

Apply the distributive property.

$2 x + 21 = \sqrt{2 x} (\sqrt{2 x} + 3) + 3 (\sqrt{2 x} + 3)$

Apply the distributive property.

$2 x + 21 = \sqrt{2 x} \sqrt{2 x} + \sqrt{2 x} \cdot 3 + 3 (\sqrt{2 x} + 3)$

Apply the distributive property.

$2 x + 21 = \sqrt{2 x} \sqrt{2 x} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{2 x} \sqrt{2 x}$ .

Raise $\sqrt{2 x}$ to the power of $1$ .

$2 x + 21 = {\sqrt{2 x}}^{1} \sqrt{2 x} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Raise $\sqrt{2 x}$ to the power of $1$ .

$2 x + 21 = {\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

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

$2 x + 21 = {\sqrt{2 x}}^{1 + 1} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Add $1$ and $1$ .

$2 x + 21 = {\sqrt{2 x}}^{2} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

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

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

$2 x + 21 = {({(2 x)}^{\frac{1}{2}})}^{2} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

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

$2 x + 21 = {(2 x)}^{\frac{1}{2} \cdot 2} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

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

$2 x + 21 = {(2 x)}^{\frac{2}{2}} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x + 21 = {(2 x)}^{\frac{2}{2}} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Rewrite the expression.

$2 x + 21 = {(2 x)}^{1} + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Simplify.

$2 x + 21 = 2 x + \sqrt{2 x} \cdot 3 + 3 \sqrt{2 x} + 3 \cdot 3$

Move $3$ to the left of $\sqrt{2 x}$ .

$2 x + 21 = 2 x + 3 \cdot \sqrt{2 x} + 3 \sqrt{2 x} + 3 \cdot 3$

Multiply $3$ by $3$ .

$2 x + 21 = 2 x + 3 \sqrt{2 x} + 3 \sqrt{2 x} + 9$

Add $3 \sqrt{2 x}$ and $3 \sqrt{2 x}$ .

$2 x + 21 = 2 x + 6 \sqrt{2 x} + 9$

Step 3

Solve for $6 \sqrt{2 x}$ .

Rewrite the equation as $2 x + 6 \sqrt{2 x} + 9 = 2 x + 21$ .

$2 x + 6 \sqrt{2 x} + 9 = 2 x + 21$

Move all terms not containing $6 \sqrt{2 x}$ to the right side of the equation.

Subtract $2 x$ from both sides of the equation.

$6 \sqrt{2 x} + 9 = 2 x + 21 - 2 x$

Subtract $9$ from both sides of the equation.

$6 \sqrt{2 x} = 2 x + 21 - 2 x - 9$

Combine the opposite terms in $2 x + 21 - 2 x - 9$ .

Subtract $2 x$ from $2 x$ .

$6 \sqrt{2 x} = 0 + 21 - 9$

Add $0$ and $21$ .

$6 \sqrt{2 x} = 21 - 9$

Subtract $9$ from $21$ .

$6 \sqrt{2 x} = 12$

Step 4

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

${(6 \sqrt{2 x})}^{2} = 12^{2}$

Step 5

Simplify each side of the equation.

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

${(6 {(2 x)}^{\frac{1}{2}})}^{2} = 12^{2}$

Simplify the left side.

Simplify ${(6 {(2 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 x$ .

${(6 (2^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = 12^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $6 \cdot 2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

${(6 \cdot 2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Apply the product rule to $6 \cdot 2^{\frac{1}{2}}$ .

$6^{2} \cdot {(2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Raise $6$ to the power of $2$ .

$36 \cdot {(2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

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

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

$36 \cdot 2^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 \cdot 2^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Rewrite the expression.

$36 \cdot 2^{1} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Evaluate the exponent.

$36 \cdot 2 {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Multiply $36$ by $2$ .

$72 {(x^{\frac{1}{2}})}^{2} = 12^{2}$

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

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

$72 x^{\frac{1}{2} \cdot 2} = 12^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$72 x^{\frac{1}{2} \cdot 2} = 12^{2}$

Rewrite the expression.

$72 x^{1} = 12^{2}$

Simplify.

$72 x = 12^{2}$

Simplify the right side.

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

$72 x = 144$

Step 6

Divide each term in $72 x = 144$ by $72$ and simplify.

Divide each term in $72 x = 144$ by $72$ .

$\frac{72 x}{72} = \frac{144}{72}$

Simplify the left side.

Cancel the common factor of $72$ .

Cancel the common factor.

$\frac{72 x}{72} = \frac{144}{72}$

Divide $x$ by $1$ .

$x = \frac{144}{72}$

Simplify the right side.

Divide $144$ by $72$ .

$x = 2$

Solve for x Solve square root of 2x+21 = square root of 2x+3

Related Questions

Support LCMGCF.com by disabling your adblocker.