Solve for x square root of 1+x square root of x^2+24=x+1

$\sqrt{1 + x \sqrt{x^{2} + 24}} = x + 1$

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

${\sqrt{1 + x \sqrt{x^{2} + 24}}}^{2} = {(x + 1)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x \sqrt{x^{2} + 24}}$ as ${(1 + x \sqrt{x^{2} + 24})}^{\frac{1}{2}}$ .

${({(1 + x \sqrt{x^{2} + 24})}^{\frac{1}{2}})}^{2} = {(x + 1)}^{2}$

Simplify the left side.

Simplify ${({(1 + x \sqrt{x^{2} + 24})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(1 + x \sqrt{x^{2} + 24})}^{\frac{1}{2}})}^{2}$ .

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

${(1 + x \sqrt{x^{2} + 24})}^{\frac{1}{2} \cdot 2} = {(x + 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(1 + x \sqrt{x^{2} + 24})}^{\frac{1}{2} \cdot 2} = {(x + 1)}^{2}$

Rewrite the expression.

${(1 + x \sqrt{x^{2} + 24})}^{1} = {(x + 1)}^{2}$

Simplify.

$1 + x \sqrt{x^{2} + 24} = {(x + 1)}^{2}$

Simplify the right side.

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

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$1 + x \sqrt{x^{2} + 24} = (x + 1) (x + 1)$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$1 + x \sqrt{x^{2} + 24} = x (x + 1) + 1 (x + 1)$

Apply the distributive property.

$1 + x \sqrt{x^{2} + 24} = x \cdot x + x \cdot 1 + 1 (x + 1)$

Apply the distributive property.

$1 + x \sqrt{x^{2} + 24} = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$1 + x \sqrt{x^{2} + 24} = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$1 + x \sqrt{x^{2} + 24} = x^{2} + x + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$1 + x \sqrt{x^{2} + 24} = x^{2} + x + x + 1 \cdot 1$

Multiply $1$ by $1$ .

$1 + x \sqrt{x^{2} + 24} = x^{2} + x + x + 1$

Add $x$ and $x$ .

$1 + x \sqrt{x^{2} + 24} = x^{2} + 2 x + 1$

Move all terms not containing $x \sqrt{x^{2} + 24}$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$x \sqrt{x^{2} + 24} = x^{2} + 2 x + 1 - 1$

Combine the opposite terms in $x^{2} + 2 x + 1 - 1$ .

Subtract $1$ from $1$ .

$x \sqrt{x^{2} + 24} = x^{2} + 2 x + 0$

Add $x^{2} + 2 x$ and $0$ .

$x \sqrt{x^{2} + 24} = x^{2} + 2 x$

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

${(x \sqrt{x^{2} + 24})}^{2} = {(x^{2} + 2 x)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 24}$ as ${(x^{2} + 24)}^{\frac{1}{2}}$ .

${(x {(x^{2} + 24)}^{\frac{1}{2}})}^{2} = {(x^{2} + 2 x)}^{2}$

Simplify the left side.

Simplify ${(x {(x^{2} + 24)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $x {(x^{2} + 24)}^{\frac{1}{2}}$ .

$x^{2} {({(x^{2} + 24)}^{\frac{1}{2}})}^{2} = {(x^{2} + 2 x)}^{2}$

Multiply the exponents in ${({(x^{2} + 24)}^{\frac{1}{2}})}^{2}$ .

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

$x^{2} {(x^{2} + 24)}^{\frac{1}{2} \cdot 2} = {(x^{2} + 2 x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} {(x^{2} + 24)}^{\frac{1}{2} \cdot 2} = {(x^{2} + 2 x)}^{2}$

Rewrite the expression.

$x^{2} {(x^{2} + 24)}^{1} = {(x^{2} + 2 x)}^{2}$

Simplify.

$x^{2} (x^{2} + 24) = {(x^{2} + 2 x)}^{2}$

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 24 = {(x^{2} + 2 x)}^{2}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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

$x^{2 + 2} + x^{2} \cdot 24 = {(x^{2} + 2 x)}^{2}$

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 24 = {(x^{2} + 2 x)}^{2}$

Move $24$ to the left of $x^{2}$ .

$x^{4} + 24 x^{2} = {(x^{2} + 2 x)}^{2}$

Simplify the right side.

Simplify ${(x^{2} + 2 x)}^{2}$ .

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

$x^{4} + 24 x^{2} = (x^{2} + 2 x) (x^{2} + 2 x)$

Expand $(x^{2} + 2 x) (x^{2} + 2 x)$ using the FOIL Method.

Apply the distributive property.

$x^{4} + 24 x^{2} = x^{2} (x^{2} + 2 x) + 2 x (x^{2} + 2 x)$

Apply the distributive property.

$x^{4} + 24 x^{2} = x^{2} x^{2} + x^{2} (2 x) + 2 x (x^{2} + 2 x)$

Apply the distributive property.

$x^{4} + 24 x^{2} = x^{2} x^{2} + x^{2} (2 x) + 2 x \cdot x^{2} + 2 x (2 x)$

Simplify and combine like terms.

Simplify each term.

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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

$x^{4} + 24 x^{2} = x^{2 + 2} + x^{2} (2 x) + 2 x \cdot x^{2} + 2 x (2 x)$

Add $2$ and $2$ .

$x^{4} + 24 x^{2} = x^{4} + x^{2} (2 x) + 2 x \cdot x^{2} + 2 x (2 x)$

Rewrite using the commutative property of multiplication.

$x^{4} + 24 x^{2} = x^{4} + 2 x^{2} x + 2 x \cdot x^{2} + 2 x (2 x)$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$x^{4} + 24 x^{2} = x^{4} + 2 (x \cdot x^{2}) + 2 x \cdot x^{2} + 2 x (2 x)$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$x^{4} + 24 x^{2} = x^{4} + 2 (x^{1} x^{2}) + 2 x \cdot x^{2} + 2 x (2 x)$

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

$x^{4} + 24 x^{2} = x^{4} + 2 x^{1 + 2} + 2 x \cdot x^{2} + 2 x (2 x)$

Add $1$ and $2$ .

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 x \cdot x^{2} + 2 x (2 x)$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 (x^{2} x) + 2 x (2 x)$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 (x^{2} x^{1}) + 2 x (2 x)$

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

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 x^{2 + 1} + 2 x (2 x)$

Add $2$ and $1$ .

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 x^{3} + 2 x (2 x)$

Rewrite using the commutative property of multiplication.

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 x^{3} + 2 \cdot 2 x \cdot x$

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

Move $x$ .

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 x^{3} + 2 \cdot 2 (x \cdot x)$

Multiply $x$ by $x$ .

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 x^{3} + 2 \cdot 2 x^{2}$

Multiply $2$ by $2$ .

$x^{4} + 24 x^{2} = x^{4} + 2 x^{3} + 2 x^{3} + 4 x^{2}$

Add $2 x^{3}$ and $2 x^{3}$ .

$x^{4} + 24 x^{2} = x^{4} + 4 x^{3} + 4 x^{2}$

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.

$x^{4} + 4 x^{3} + 4 x^{2} = x^{4} + 24 x^{2}$

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

Subtract $x^{4}$ from both sides of the equation.

$x^{4} + 4 x^{3} + 4 x^{2} - x^{4} = 24 x^{2}$

Subtract $24 x^{2}$ from both sides of the equation.

$x^{4} + 4 x^{3} + 4 x^{2} - x^{4} - 24 x^{2} = 0$

Combine the opposite terms in $x^{4} + 4 x^{3} + 4 x^{2} - x^{4} - 24 x^{2}$ .

Subtract $x^{4}$ from $x^{4}$ .

$4 x^{3} + 4 x^{2} + 0 - 24 x^{2} = 0$

Add $4 x^{3} + 4 x^{2}$ and $0$ .

$4 x^{3} + 4 x^{2} - 24 x^{2} = 0$

Subtract $24 x^{2}$ from $4 x^{2}$ .

$4 x^{3} - 20 x^{2} = 0$

Factor $4 x^{2}$ out of $4 x^{3} - 20 x^{2}$ .

Factor $4 x^{2}$ out of $4 x^{3}$ .

$4 x^{2} (x) - 20 x^{2} = 0$

Factor $4 x^{2}$ out of $- 20 x^{2}$ .

$4 x^{2} (x) + 4 x^{2} (- 5) = 0$

Factor $4 x^{2}$ out of $4 x^{2} (x) + 4 x^{2} (- 5)$ .

$4 x^{2} (x - 5) = 0$

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

$x^{2} = 0$

$x - 5 = 0$

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $4 x^{2} (x - 5) = 0$ true.

$x = 0, 5$

Solve for x square root of 1+x square root of x^2+24=x+1

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