Solve sqrt{x^2}-1=4^2-x^2

$$\sqrt{ { x }^{ 2 } } -1 = { 4 }^{ 2 } - { x }^{ 2 }$$

Solve for x

$$x = \frac{\sqrt{69} - 1}{2} \approx 3.653311931$$

$$x=\frac{1-\sqrt{69}}{2}\approx -3.653311931$$

Solution Steps

Subtract -1 from both sides of the equation.

$$\sqrt{x^{2}}=4^{2}-x^{2}+1$$

Calculate 4 to the power of 2 and get 16.

$$\sqrt{x^{2}}=16-x^{2}+1$$

Add 16 and 1 to get 17.

$$\sqrt{x^{2}}=17-x^{2}$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}}\right)^{2}=\left(17-x^{2}\right)^{2}$$

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

$$x^{2}=\left(17-x^{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(17-x^{2}\right)^{2}.

$$x^{2}=289-34x^{2}+\left(x^{2}\right)^{2}$$

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

$$x^{2}=289-34x^{2}+x^{4}$$

Subtract 289 from both sides.

$$x^{2}-289=-34x^{2}+x^{4}$$

Add 34x^{2} to both sides.

$$x^{2}-289+34x^{2}=x^{4}$$

Combine x^{2} and 34x^{2} to get 35x^{2}.

$$35x^{2}-289=x^{4}$$

Subtract x^{4} from both sides.

$$35x^{2}-289-x^{4}=0$$

Substitute t for x^{2}.

$$-t^{2}+35t-289=0$$

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, 35 for b, and -289 for c in the quadratic formula.

$$t=\frac{-35±\sqrt{35^{2}-4\left(-1\right)\left(-289\right)}}{-2}$$

Do the calculations.

$$t=\frac{-35±\sqrt{69}}{-2}$$

Solve the equation t=\frac{-35±\sqrt{69}}{-2} when ± is plus and when ± is minus.

$$t=\frac{35-\sqrt{69}}{2} t=\frac{\sqrt{69}+35}{2}$$

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

$$x=-\frac{1-\sqrt{69}}{2} x=\frac{1-\sqrt{69}}{2} x=\frac{\sqrt{69}+1}{2} x=-\frac{\sqrt{69}+1}{2}$$

Substitute -\frac{1-\sqrt{69}}{2} for x in the equation \sqrt{x^{2}}-1=4^{2}-x^{2}.

$$\sqrt{\left(-\frac{1-\sqrt{69}}{2}\right)^{2}}-1=4^{2}-\left(-\frac{1-\sqrt{69}}{2}\right)^{2}$$

Simplify. The value x=-\frac{1-\sqrt{69}}{2} satisfies the equation.

$$-\frac{3}{2}+\frac{1}{2}\times 69^{\frac{1}{2}}=-\frac{3}{2}+\frac{1}{2}\times 69^{\frac{1}{2}}$$

Substitute \frac{1-\sqrt{69}}{2} for x in the equation \sqrt{x^{2}}-1=4^{2}-x^{2}.

$$\sqrt{\left(\frac{1-\sqrt{69}}{2}\right)^{2}}-1=4^{2}-\left(\frac{1-\sqrt{69}}{2}\right)^{2}$$

Simplify. The value x=\frac{1-\sqrt{69}}{2} satisfies the equation.

$$-\frac{3}{2}+\frac{1}{2}\times 69^{\frac{1}{2}}=-\frac{3}{2}+\frac{1}{2}\times 69^{\frac{1}{2}}$$

Substitute \frac{\sqrt{69}+1}{2} for x in the equation \sqrt{x^{2}}-1=4^{2}-x^{2}.

$$\sqrt{\left(\frac{\sqrt{69}+1}{2}\right)^{2}}-1=4^{2}-\left(\frac{\sqrt{69}+1}{2}\right)^{2}$$

Simplify. The value x=\frac{\sqrt{69}+1}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{1}{2}+\frac{1}{2}\times 69^{\frac{1}{2}}=-\frac{3}{2}-\frac{1}{2}\times 69^{\frac{1}{2}}$$

Substitute -\frac{\sqrt{69}+1}{2} for x in the equation \sqrt{x^{2}}-1=4^{2}-x^{2}.

$$\sqrt{\left(-\frac{\sqrt{69}+1}{2}\right)^{2}}-1=4^{2}-\left(-\frac{\sqrt{69}+1}{2}\right)^{2}$$

Simplify. The value x=-\frac{\sqrt{69}+1}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{1}{2}+\frac{1}{2}\times 69^{\frac{1}{2}}=-\frac{3}{2}-\frac{1}{2}\times 69^{\frac{1}{2}}$$

List all solutions of \sqrt{x^{2}}=17-x^{2}.

$$x=-\frac{1-\sqrt{69}}{2} x=\frac{1-\sqrt{69}}{2}$$