Solve sqrt{z^2+8}=1-2z

$$\sqrt { z ^ { 2 } + 8 } = 1 - 2 z$$

Solve for z

$$z=-1$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{z^{2}+8}\right)^{2}=\left(1-2z\right)^{2}$$

Calculate \sqrt{z^{2}+8} to the power of 2 and get z^{2}+8.

$$z^{2}+8=\left(1-2z\right)^{2}$$

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

$$z^{2}+8=1-4z+4z^{2}$$

Subtract 1 from both sides.

$$z^{2}+8-1=-4z+4z^{2}$$

Subtract 1 from 8 to get 7.

$$z^{2}+7=-4z+4z^{2}$$

Add 4z to both sides.

$$z^{2}+7+4z=4z^{2}$$

Subtract 4z^{2} from both sides.

$$z^{2}+7+4z-4z^{2}=0$$

Combine z^{2} and -4z^{2} to get -3z^{2}.

$$-3z^{2}+7+4z=0$$

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

$$-3z^{2}+4z+7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3z^{2}+az+bz+7. To find a and b, set up a system to be solved.

$$a+b=4 ab=-3\times 7=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 4.

$$a=7 b=-3$$

Rewrite -3z^{2}+4z+7 as \left(-3z^{2}+7z\right)+\left(-3z+7\right).

$$\left(-3z^{2}+7z\right)+\left(-3z+7\right)$$

Factor out -z in the first and -1 in the second group.

$$-z\left(3z-7\right)-\left(3z-7\right)$$

Factor out common term 3z-7 by using distributive property.

$$\left(3z-7\right)\left(-z-1\right)$$

To find equation solutions, solve 3z-7=0 and -z-1=0.

$$z=\frac{7}{3} z=-1$$

Substitute \frac{7}{3} for z in the equation \sqrt{z^{2}+8}=1-2z.

$$\sqrt{\left(\frac{7}{3}\right)^{2}+8}=1-2\times \frac{7}{3}$$

Simplify. The value z=\frac{7}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{11}{3}=-\frac{11}{3}$$

Substitute -1 for z in the equation \sqrt{z^{2}+8}=1-2z.

$$\sqrt{\left(-1\right)^{2}+8}=1-2\left(-1\right)$$

Simplify. The value z=-1 satisfies the equation.

$$3=3$$

Equation \sqrt{z^{2}+8}=1-2z has a unique solution.

$$z=-1$$