Solve for x square root of x+5 = square root of 8x+28

$\sqrt{x} + 5 = \sqrt{8 x + 28}$

Step 1

Solve for $\sqrt{x}$ .

Simplify $\sqrt{8 x + 28}$ .

Rewrite.

$\sqrt{x} + 5 = 0 + 0 + \sqrt{8 x + 28}$

Simplify by adding zeros.

$\sqrt{x} + 5 = \sqrt{8 x + 28}$

Factor $4$ out of $8 x + 28$ .

Factor $4$ out of $8 x$ .

$\sqrt{x} + 5 = \sqrt{4 (2 x) + 28}$

Factor $4$ out of $28$ .

$\sqrt{x} + 5 = \sqrt{4 (2 x) + 4 (7)}$

Factor $4$ out of $4 (2 x) + 4 (7)$ .

$\sqrt{x} + 5 = \sqrt{4 (2 x + 7)}$

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

$\sqrt{x} + 5 = \sqrt{2^{2} (2 x + 7)}$

Pull terms out from under the radical.

$\sqrt{x} + 5 = 2 \sqrt{2 x + 7}$

Subtract $5$ from both sides of the equation.

$\sqrt{x} = 2 \sqrt{2 x + 7} - 5$

Step 2

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

${\sqrt{x}}^{2} = {(2 \sqrt{2 x + 7} - 5)}^{2}$

Step 3

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(2 \sqrt{2 x + 7} - 5)}^{2}$

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = {(2 \sqrt{2 x + 7} - 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(2 \sqrt{2 x + 7} - 5)}^{2}$

Rewrite the expression.

$x^{1} = {(2 \sqrt{2 x + 7} - 5)}^{2}$

Simplify.

$x = {(2 \sqrt{2 x + 7} - 5)}^{2}$

Simplify the right side.

Simplify ${(2 \sqrt{2 x + 7} - 5)}^{2}$ .

Rewrite ${(2 \sqrt{2 x + 7} - 5)}^{2}$ as $(2 \sqrt{2 x + 7} - 5) (2 \sqrt{2 x + 7} - 5)$ .

$x = (2 \sqrt{2 x + 7} - 5) (2 \sqrt{2 x + 7} - 5)$

Expand $(2 \sqrt{2 x + 7} - 5) (2 \sqrt{2 x + 7} - 5)$ using the FOIL Method.

Apply the distributive property.

$x = 2 \sqrt{2 x + 7} (2 \sqrt{2 x + 7} - 5) - 5 (2 \sqrt{2 x + 7} - 5)$

Apply the distributive property.

$x = 2 \sqrt{2 x + 7} (2 \sqrt{2 x + 7}) + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7} - 5)$

Apply the distributive property.

$x = 2 \sqrt{2 x + 7} (2 \sqrt{2 x + 7}) + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

Multiply $2 \sqrt{2 x + 7} (2 \sqrt{2 x + 7})$ .

Multiply $2$ by $2$ .

$x = 4 \sqrt{2 x + 7} \sqrt{2 x + 7} + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Raise $\sqrt{2 x + 7}$ to the power of $1$ .

$x = 4 ({\sqrt{2 x + 7}}^{1} \sqrt{2 x + 7}) + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Raise $\sqrt{2 x + 7}$ to the power of $1$ .

$x = 4 ({\sqrt{2 x + 7}}^{1} {\sqrt{2 x + 7}}^{1}) + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

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

$x = 4 {\sqrt{2 x + 7}}^{1 + 1} + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Add $1$ and $1$ .

$x = 4 {\sqrt{2 x + 7}}^{2} + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Rewrite ${\sqrt{2 x + 7}}^{2}$ as $2 x + 7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 7}$ as ${(2 x + 7)}^{\frac{1}{2}}$ .

$x = 4 {({(2 x + 7)}^{\frac{1}{2}})}^{2} + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

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

$x = 4 {(2 x + 7)}^{\frac{1}{2} \cdot 2} + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

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

$x = 4 {(2 x + 7)}^{\frac{2}{2}} + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 4 {(2 x + 7)}^{\frac{2}{2}} + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Rewrite the expression.

$x = 4 {(2 x + 7)}^{1} + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Simplify.

$x = 4 (2 x + 7) + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Apply the distributive property.

$x = 4 (2 x) + 4 \cdot 7 + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Multiply $2$ by $4$ .

$x = 8 x + 4 \cdot 7 + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Multiply $4$ by $7$ .

$x = 8 x + 28 + 2 \sqrt{2 x + 7} \cdot - 5 - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Multiply $- 5$ by $2$ .

$x = 8 x + 28 - 10 \sqrt{2 x + 7} - 5 (2 \sqrt{2 x + 7}) - 5 \cdot - 5$

Multiply $2$ by $- 5$ .

$x = 8 x + 28 - 10 \sqrt{2 x + 7} - 10 \sqrt{2 x + 7} - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$x = 8 x + 28 - 10 \sqrt{2 x + 7} - 10 \sqrt{2 x + 7} + 25$

Add $28$ and $25$ .

$x = 8 x + 53 - 10 \sqrt{2 x + 7} - 10 \sqrt{2 x + 7}$

Subtract $10 \sqrt{2 x + 7}$ from $- 10 \sqrt{2 x + 7}$ .

$x = 8 x + 53 - 20 \sqrt{2 x + 7}$

Step 4

Solve for $- 20 \sqrt{2 x + 7}$ .

Rewrite the equation as $8 x + 53 - 20 \sqrt{2 x + 7} = x$ .

$8 x + 53 - 20 \sqrt{2 x + 7} = x$

Move all terms not containing $- 20 \sqrt{2 x + 7}$ to the right side of the equation.

Subtract $8 x$ from both sides of the equation.

$53 - 20 \sqrt{2 x + 7} = x - 8 x$

Subtract $53$ from both sides of the equation.

$- 20 \sqrt{2 x + 7} = x - 8 x - 53$

Subtract $8 x$ from $x$ .

$- 20 \sqrt{2 x + 7} = - 7 x - 53$

Step 5

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

${(- 20 \sqrt{2 x + 7})}^{2} = {(- 7 x - 53)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 7}$ as ${(2 x + 7)}^{\frac{1}{2}}$ .

${(- 20 {(2 x + 7)}^{\frac{1}{2}})}^{2} = {(- 7 x - 53)}^{2}$

Simplify the left side.

Simplify ${(- 20 {(2 x + 7)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 20 {(2 x + 7)}^{\frac{1}{2}}$ .

${(- 20)}^{2} {({(2 x + 7)}^{\frac{1}{2}})}^{2} = {(- 7 x - 53)}^{2}$

Raise $- 20$ to the power of $2$ .

$400 {({(2 x + 7)}^{\frac{1}{2}})}^{2} = {(- 7 x - 53)}^{2}$

Multiply the exponents in ${({(2 x + 7)}^{\frac{1}{2}})}^{2}$ .

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

$400 {(2 x + 7)}^{\frac{1}{2} \cdot 2} = {(- 7 x - 53)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$400 {(2 x + 7)}^{\frac{1}{2} \cdot 2} = {(- 7 x - 53)}^{2}$

Rewrite the expression.

$400 {(2 x + 7)}^{1} = {(- 7 x - 53)}^{2}$

Simplify.

$400 (2 x + 7) = {(- 7 x - 53)}^{2}$

Apply the distributive property.

$400 (2 x) + 400 \cdot 7 = {(- 7 x - 53)}^{2}$

Multiply.

Multiply $2$ by $400$ .

$800 x + 400 \cdot 7 = {(- 7 x - 53)}^{2}$

Multiply $400$ by $7$ .

$800 x + 2800 = {(- 7 x - 53)}^{2}$

Simplify the right side.

Simplify ${(- 7 x - 53)}^{2}$ .

Rewrite ${(- 7 x - 53)}^{2}$ as $(- 7 x - 53) (- 7 x - 53)$ .

$800 x + 2800 = (- 7 x - 53) (- 7 x - 53)$

Expand $(- 7 x - 53) (- 7 x - 53)$ using the FOIL Method.

Apply the distributive property.

$800 x + 2800 = - 7 x (- 7 x - 53) - 53 (- 7 x - 53)$

Apply the distributive property.

$800 x + 2800 = - 7 x (- 7 x) - 7 x \cdot - 53 - 53 (- 7 x - 53)$

Apply the distributive property.

$800 x + 2800 = - 7 x (- 7 x) - 7 x \cdot - 53 - 53 (- 7 x) - 53 \cdot - 53$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$800 x + 2800 = - 7 \cdot - 7 x \cdot x - 7 x \cdot - 53 - 53 (- 7 x) - 53 \cdot - 53$

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

Move $x$ .

$800 x + 2800 = - 7 \cdot - 7 (x \cdot x) - 7 x \cdot - 53 - 53 (- 7 x) - 53 \cdot - 53$

Multiply $x$ by $x$ .

$800 x + 2800 = - 7 \cdot - 7 x^{2} - 7 x \cdot - 53 - 53 (- 7 x) - 53 \cdot - 53$

Multiply $- 7$ by $- 7$ .

$800 x + 2800 = 49 x^{2} - 7 x \cdot - 53 - 53 (- 7 x) - 53 \cdot - 53$

Multiply $- 53$ by $- 7$ .

$800 x + 2800 = 49 x^{2} + 371 x - 53 (- 7 x) - 53 \cdot - 53$

Multiply $- 7$ by $- 53$ .

$800 x + 2800 = 49 x^{2} + 371 x + 371 x - 53 \cdot - 53$

Multiply $- 53$ by $- 53$ .

$800 x + 2800 = 49 x^{2} + 371 x + 371 x + 2809$

Add $371 x$ and $371 x$ .

$800 x + 2800 = 49 x^{2} + 742 x + 2809$

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.

$49 x^{2} + 742 x + 2809 = 800 x + 2800$

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

Subtract $800 x$ from both sides of the equation.

$49 x^{2} + 742 x + 2809 - 800 x = 2800$

Subtract $800 x$ from $742 x$ .

$49 x^{2} - 58 x + 2809 = 2800$

Subtract $2800$ from both sides of the equation.

$49 x^{2} - 58 x + 2809 - 2800 = 0$

Subtract $2800$ from $2809$ .

$49 x^{2} - 58 x + 9 = 0$

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 $a \cdot c = 49 \cdot 9 = 441$ and whose sum is $b = - 58$ .

Factor $- 58$ out of $- 58 x$ .

$49 x^{2} - 58 x + 9 = 0$

Rewrite $- 58$ as $- 9$ plus $- 49$

$49 x^{2} + (- 9 - 49) x + 9 = 0$

Apply the distributive property.

$49 x^{2} - 9 x - 49 x + 9 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(49 x^{2} - 9 x) - 49 x + 9 = 0$

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

$x (49 x - 9) - (49 x - 9) = 0$

Factor the polynomial by factoring out the greatest common factor, $49 x - 9$ .

$(49 x - 9) (x - 1) = 0$

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

$49 x - 9 = 0$

$x - 1 = 0$

Set $49 x - 9$ equal to $0$ and solve for $x$ .

Set $49 x - 9$ equal to $0$ .

$49 x - 9 = 0$

Solve $49 x - 9 = 0$ for $x$ .

Add $9$ to both sides of the equation.

$49 x = 9$

Divide each term in $49 x = 9$ by $49$ and simplify.

Divide each term in $49 x = 9$ by $49$ .

$\frac{49 x}{49} = \frac{9}{49}$

Simplify the left side.

Cancel the common factor of $49$ .

Cancel the common factor.

$\frac{49 x}{49} = \frac{9}{49}$

Divide $x$ by $1$ .

$x = \frac{9}{49}$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(49 x - 9) (x - 1) = 0$ true.

$x = \frac{9}{49}, 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{49}, 1$

Decimal Form:

$x = 0.18367346 \dots, 1$

Solve for x square root of x+5 = square root of 8x+28

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