Solve frac{sqrt{5-x^2}}{x+1}=1

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

Solve for x

$$x=1$$

Solution Steps

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

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

Square both sides of the equation.

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

Calculate \sqrt{5-x^{2}} to the power of 2 and get 5-x^{2}.

$$5-x^{2}=\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$5-x^{2}=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$5-x^{2}-x^{2}=2x+1$$

Combine -x^{2} and -x^{2} to get -2x^{2}.

$$5-2x^{2}=2x+1$$

Subtract 2x from both sides.

$$5-2x^{2}-2x=1$$

Subtract 1 from both sides.

$$5-2x^{2}-2x-1=0$$

Subtract 1 from 5 to get 4.

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

Divide both sides by 2.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=-1 ab=-2=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=1 b=-2$$

Rewrite -x^{2}-x+2 as \left(-x^{2}+x\right)+\left(-2x+2\right).

$$\left(-x^{2}+x\right)+\left(-2x+2\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(-x+1\right)+2\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(x+2\right)$$

To find equation solutions, solve -x+1=0 and x+2=0.

$$x=1 x=-2$$

Substitute 1 for x in the equation \frac{\sqrt{5-x^{2}}}{x+1}=1.

$$\frac{\sqrt{5-1^{2}}}{1+1}=1$$

Simplify. The value x=1 satisfies the equation.

$$1=1$$

Substitute -2 for x in the equation \frac{\sqrt{5-x^{2}}}{x+1}=1.

$$\frac{\sqrt{5-\left(-2\right)^{2}}}{-2+1}=1$$

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Equation \sqrt{5-x^{2}}=x+1 has a unique solution.

$$x=1$$