Solve for x square root of 25-x^2=x+1

$\sqrt{25 - x^{2}} = x + 1$

Step 1

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

${\sqrt{25 - x^{2}}}^{2} = {(x + 1)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 - x^{2}}$ as ${(25 - x^{2})}^{\frac{1}{2}}$ .

${({(25 - x^{2})}^{\frac{1}{2}})}^{2} = {(x + 1)}^{2}$

Simplify the left side.

Simplify ${({(25 - x^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(25 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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

${(25 - x^{2})}^{\frac{1}{2} \cdot 2} = {(x + 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(25 - x^{2})}^{\frac{1}{2} \cdot 2} = {(x + 1)}^{2}$

Rewrite the expression.

${(25 - x^{2})}^{1} = {(x + 1)}^{2}$

Simplify.

$25 - x^{2} = {(x + 1)}^{2}$

Simplify the right side.

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

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

$25 - x^{2} = (x + 1) (x + 1)$

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

Apply the distributive property.

$25 - x^{2} = x (x + 1) + 1 (x + 1)$

Apply the distributive property.

$25 - x^{2} = x \cdot x + x \cdot 1 + 1 (x + 1)$

Apply the distributive property.

$25 - x^{2} = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$25 - x^{2} = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$25 - x^{2} = x^{2} + x + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$25 - x^{2} = x^{2} + x + x + 1 \cdot 1$

Multiply $1$ by $1$ .

$25 - x^{2} = x^{2} + x + x + 1$

Add $x$ and $x$ .

$25 - x^{2} = x^{2} + 2 x + 1$

Step 3

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.

$x^{2} + 2 x + 1 = 25 - x^{2}$

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

Add $x^{2}$ to both sides of the equation.

$x^{2} + 2 x + 1 + x^{2} = 25$

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

$2 x^{2} + 2 x + 1 = 25$

Subtract $25$ from both sides of the equation.

$2 x^{2} + 2 x + 1 - 25 = 0$

Subtract $25$ from $1$ .

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

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} + 2 x - 24$ .

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

$2 (x^{2}) + 2 x - 24 = 0$

Factor $2$ out of $2 x$ .

$2 (x^{2}) + 2 (x) - 24 = 0$

Factor $2$ out of $- 24$ .

$2 (x^{2}) + 2 x + 2 \cdot - 12 = 0$

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

$2 (x^{2} + x) + 2 \cdot - 12 = 0$

Factor $2$ out of $2 (x^{2} + x) + 2 \cdot - 12$ .

$2 (x^{2} + x - 12) = 0$

Factor.

Factor $x^{2} + x - 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

$2 ((x - 3) (x + 4)) = 0$

Remove unnecessary parentheses.

$2 (x - 3) (x + 4) = 0$

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

$x - 3 = 0$

$x + 4 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $2 (x - 3) (x + 4) = 0$ true.

$x = 3, - 4$

Step 4

Exclude the solutions that do not make $\sqrt{25 - x^{2}} = x + 1$ true.

$x = 3$

Solve for x square root of 25-x^2=x+1

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