Solve x^2+11+sqrt{x^2+11}=42

$$x ^ { 2 } + 11 + \sqrt { x ^ { 2 } + 11 } = 42$$

Solve for x

$$x=-5$$

$$x=5$$

Solution Steps

Subtract x^{2}+11 from both sides of the equation.

$$\sqrt{x^{2}+11}=42-\left(x^{2}+11\right)$$

To find the opposite of x^{2}+11, find the opposite of each term.

$$\sqrt{x^{2}+11}=42-x^{2}-11$$

Subtract 11 from 42 to get 31.

$$\sqrt{x^{2}+11}=31-x^{2}$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}+11}\right)^{2}=\left(31-x^{2}\right)^{2}$$

Calculate \sqrt{x^{2}+11} to the power of 2 and get x^{2}+11.

$$x^{2}+11=\left(31-x^{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(31-x^{2}\right)^{2}.

$$x^{2}+11=961-62x^{2}+\left(x^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{2}+11=961-62x^{2}+x^{4}$$

Subtract 961 from both sides.

$$x^{2}+11-961=-62x^{2}+x^{4}$$

Subtract 961 from 11 to get -950.

$$x^{2}-950=-62x^{2}+x^{4}$$

Add 62x^{2} to both sides.

$$x^{2}-950+62x^{2}=x^{4}$$

Combine x^{2} and 62x^{2} to get 63x^{2}.

$$63x^{2}-950=x^{4}$$

Subtract x^{4} from both sides.

$$63x^{2}-950-x^{4}=0$$

Substitute t for x^{2}.

$$-t^{2}+63t-950=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -1 for a, 63 for b, and -950 for c in the quadratic formula.

$$t=\frac{-63±\sqrt{63^{2}-4\left(-1\right)\left(-950\right)}}{-2}$$

Do the calculations.

$$t=\frac{-63±13}{-2}$$

Solve the equation t=\frac{-63±13}{-2} when ± is plus and when ± is minus.

$$t=25 t=38$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=5 x=-5 x=\sqrt{38} x=-\sqrt{38}$$

Substitute 5 for x in the equation x^{2}+11+\sqrt{x^{2}+11}=42.

$$5^{2}+11+\sqrt{5^{2}+11}=42$$

Simplify. The value x=5 satisfies the equation.

$$42=42$$

Substitute -5 for x in the equation x^{2}+11+\sqrt{x^{2}+11}=42.

$$\left(-5\right)^{2}+11+\sqrt{\left(-5\right)^{2}+11}=42$$

Simplify. The value x=-5 satisfies the equation.

$$42=42$$

Substitute \sqrt{38} for x in the equation x^{2}+11+\sqrt{x^{2}+11}=42.

$$\left(\sqrt{38}\right)^{2}+11+\sqrt{\left(\sqrt{38}\right)^{2}+11}=42$$

Simplify. The value x=\sqrt{38} does not satisfy the equation.

$$56=42$$

Substitute -\sqrt{38} for x in the equation x^{2}+11+\sqrt{x^{2}+11}=42.

$$\left(-\sqrt{38}\right)^{2}+11+\sqrt{\left(-\sqrt{38}\right)^{2}+11}=42$$

Simplify. The value x=-\sqrt{38} does not satisfy the equation.

$$56=42$$

List all solutions of \sqrt{x^{2}+11}=31-x^{2}.

$$x=5 x=-5$$