Solve for x square root of 8x+2 = square root of 10x+11

$\sqrt{8 x} + 2 = \sqrt{10 x + 11}$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{8 x} = \sqrt{10 x + 11} - 2$

Step 2

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

${\sqrt{8 x}}^{2} = {(\sqrt{10 x + 11} - 2)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x}$ as ${(8 x)}^{\frac{1}{2}}$ .

${({(8 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{10 x + 11} - 2)}^{2}$

Simplify the left side.

Simplify ${({(8 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(8 x)}^{\frac{1}{2}})}^{2}$ .

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

${(8 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{10 x + 11} - 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(8 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{10 x + 11} - 2)}^{2}$

Rewrite the expression.

${(8 x)}^{1} = {(\sqrt{10 x + 11} - 2)}^{2}$

Simplify.

$8 x = {(\sqrt{10 x + 11} - 2)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{10 x + 11} - 2)}^{2}$ .

Rewrite ${(\sqrt{10 x + 11} - 2)}^{2}$ as $(\sqrt{10 x + 11} - 2) (\sqrt{10 x + 11} - 2)$ .

$8 x = (\sqrt{10 x + 11} - 2) (\sqrt{10 x + 11} - 2)$

Expand $(\sqrt{10 x + 11} - 2) (\sqrt{10 x + 11} - 2)$ using the FOIL Method.

Apply the distributive property.

$8 x = \sqrt{10 x + 11} (\sqrt{10 x + 11} - 2) - 2 (\sqrt{10 x + 11} - 2)$

Apply the distributive property.

$8 x = \sqrt{10 x + 11} \sqrt{10 x + 11} + \sqrt{10 x + 11} \cdot - 2 - 2 (\sqrt{10 x + 11} - 2)$

Apply the distributive property.

$8 x = \sqrt{10 x + 11} \sqrt{10 x + 11} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{10 x + 11} \sqrt{10 x + 11}$ .

Raise $\sqrt{10 x + 11}$ to the power of $1$ .

$8 x = {\sqrt{10 x + 11}}^{1} \sqrt{10 x + 11} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Raise $\sqrt{10 x + 11}$ to the power of $1$ .

$8 x = {\sqrt{10 x + 11}}^{1} {\sqrt{10 x + 11}}^{1} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

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

$8 x = {\sqrt{10 x + 11}}^{1 + 1} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Add $1$ and $1$ .

$8 x = {\sqrt{10 x + 11}}^{2} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Rewrite ${\sqrt{10 x + 11}}^{2}$ as $10 x + 11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 x + 11}$ as ${(10 x + 11)}^{\frac{1}{2}}$ .

$8 x = {({(10 x + 11)}^{\frac{1}{2}})}^{2} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

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

$8 x = {(10 x + 11)}^{\frac{1}{2} \cdot 2} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

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

$8 x = {(10 x + 11)}^{\frac{2}{2}} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$8 x = {(10 x + 11)}^{\frac{2}{2}} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Rewrite the expression.

$8 x = {(10 x + 11)}^{1} + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Simplify.

$8 x = 10 x + 11 + \sqrt{10 x + 11} \cdot - 2 - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Move $- 2$ to the left of $\sqrt{10 x + 11}$ .

$8 x = 10 x + 11 - 2 \cdot \sqrt{10 x + 11} - 2 \sqrt{10 x + 11} - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$8 x = 10 x + 11 - 2 \sqrt{10 x + 11} - 2 \sqrt{10 x + 11} + 4$

Add $11$ and $4$ .

$8 x = 10 x + 15 - 2 \sqrt{10 x + 11} - 2 \sqrt{10 x + 11}$

Subtract $2 \sqrt{10 x + 11}$ from $- 2 \sqrt{10 x + 11}$ .

$8 x = 10 x + 15 - 4 \sqrt{10 x + 11}$

Step 4

Solve for $- 4 \sqrt{10 x + 11}$ .

Rewrite the equation as $10 x + 15 - 4 \sqrt{10 x + 11} = 8 x$ .

$10 x + 15 - 4 \sqrt{10 x + 11} = 8 x$

Move all terms not containing $- 4 \sqrt{10 x + 11}$ to the right side of the equation.

Subtract $10 x$ from both sides of the equation.

$15 - 4 \sqrt{10 x + 11} = 8 x - 10 x$

Subtract $15$ from both sides of the equation.

$- 4 \sqrt{10 x + 11} = 8 x - 10 x - 15$

Subtract $10 x$ from $8 x$ .

$- 4 \sqrt{10 x + 11} = - 2 x - 15$

Step 5

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

${(- 4 \sqrt{10 x + 11})}^{2} = {(- 2 x - 15)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 x + 11}$ as ${(10 x + 11)}^{\frac{1}{2}}$ .

${(- 4 {(10 x + 11)}^{\frac{1}{2}})}^{2} = {(- 2 x - 15)}^{2}$

Simplify the left side.

Simplify ${(- 4 {(10 x + 11)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 4 {(10 x + 11)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(10 x + 11)}^{\frac{1}{2}})}^{2} = {(- 2 x - 15)}^{2}$

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

$16 {({(10 x + 11)}^{\frac{1}{2}})}^{2} = {(- 2 x - 15)}^{2}$

Multiply the exponents in ${({(10 x + 11)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(10 x + 11)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 15)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(10 x + 11)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 15)}^{2}$

Rewrite the expression.

$16 {(10 x + 11)}^{1} = {(- 2 x - 15)}^{2}$

Simplify.

$16 (10 x + 11) = {(- 2 x - 15)}^{2}$

Apply the distributive property.

$16 (10 x) + 16 \cdot 11 = {(- 2 x - 15)}^{2}$

Multiply.

Multiply $10$ by $16$ .

$160 x + 16 \cdot 11 = {(- 2 x - 15)}^{2}$

Multiply $16$ by $11$ .

$160 x + 176 = {(- 2 x - 15)}^{2}$

Simplify the right side.

Simplify ${(- 2 x - 15)}^{2}$ .

Rewrite ${(- 2 x - 15)}^{2}$ as $(- 2 x - 15) (- 2 x - 15)$ .

$160 x + 176 = (- 2 x - 15) (- 2 x - 15)$

Expand $(- 2 x - 15) (- 2 x - 15)$ using the FOIL Method.

Apply the distributive property.

$160 x + 176 = - 2 x (- 2 x - 15) - 15 (- 2 x - 15)$

Apply the distributive property.

$160 x + 176 = - 2 x (- 2 x) - 2 x \cdot - 15 - 15 (- 2 x - 15)$

Apply the distributive property.

$160 x + 176 = - 2 x (- 2 x) - 2 x \cdot - 15 - 15 (- 2 x) - 15 \cdot - 15$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$160 x + 176 = - 2 \cdot - 2 x \cdot x - 2 x \cdot - 15 - 15 (- 2 x) - 15 \cdot - 15$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$160 x + 176 = - 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 15 - 15 (- 2 x) - 15 \cdot - 15$

Multiply $x$ by $x$ .

$160 x + 176 = - 2 \cdot - 2 x^{2} - 2 x \cdot - 15 - 15 (- 2 x) - 15 \cdot - 15$

Multiply $- 2$ by $- 2$ .

$160 x + 176 = 4 x^{2} - 2 x \cdot - 15 - 15 (- 2 x) - 15 \cdot - 15$

Multiply $- 15$ by $- 2$ .

$160 x + 176 = 4 x^{2} + 30 x - 15 (- 2 x) - 15 \cdot - 15$

Multiply $- 2$ by $- 15$ .

$160 x + 176 = 4 x^{2} + 30 x + 30 x - 15 \cdot - 15$

Multiply $- 15$ by $- 15$ .

$160 x + 176 = 4 x^{2} + 30 x + 30 x + 225$

Add $30 x$ and $30 x$ .

$160 x + 176 = 4 x^{2} + 60 x + 225$

Step 7

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} + 60 x + 225 = 160 x + 176$

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

Subtract $160 x$ from both sides of the equation.

$4 x^{2} + 60 x + 225 - 160 x = 176$

Subtract $160 x$ from $60 x$ .

$4 x^{2} - 100 x + 225 = 176$

Subtract $176$ from both sides of the equation.

$4 x^{2} - 100 x + 225 - 176 = 0$

Subtract $176$ from $225$ .

$4 x^{2} - 100 x + 49 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 49 = 196$ and whose sum is $b = - 100$ .

Factor $- 100$ out of $- 100 x$ .

$4 x^{2} - 100 x + 49 = 0$

Rewrite $- 100$ as $- 2$ plus $- 98$

$4 x^{2} + (- 2 - 98) x + 49 = 0$

Apply the distributive property.

$4 x^{2} - 2 x - 98 x + 49 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} - 2 x) - 98 x + 49 = 0$

Factor out the greatest common factor (GCF) from each group.

$2 x (2 x - 1) - 49 (2 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(2 x - 1) (2 x - 49) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 1 = 0$

$2 x - 49 = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Set $2 x - 49$ equal to $0$ and solve for $x$ .

Set $2 x - 49$ equal to $0$ .

$2 x - 49 = 0$

Solve $2 x - 49 = 0$ for $x$ .

Add $49$ to both sides of the equation.

$2 x = 49$

Divide each term in $2 x = 49$ by $2$ and simplify.

Divide each term in $2 x = 49$ by $2$ .

$\frac{2 x}{2} = \frac{49}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{49}{2}$

Divide $x$ by $1$ .

$x = \frac{49}{2}$

The final solution is all the values that make $(2 x - 1) (2 x - 49) = 0$ true.

$x = \frac{1}{2}, \frac{49}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, \frac{49}{2}$

Decimal Form:

$x = 0.5, 24.5$

Mixed Number Form:

$x = \frac{1}{2}, 24 \frac{1}{2}$

Solve for x square root of 8x+2 = square root of 10x+11

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