Solve sqrt{4z+9}=z+1

$$\sqrt { 4 z + 9 } = z + 1$$

Solve for z

$$z=4$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4z+9}\right)^{2}=\left(z+1\right)^{2}$$

Calculate \sqrt{4z+9} to the power of 2 and get 4z+9.

$$4z+9=\left(z+1\right)^{2}$$

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

$$4z+9=z^{2}+2z+1$$

Subtract z^{2} from both sides.

$$4z+9-z^{2}=2z+1$$

Subtract 2z from both sides.

$$4z+9-z^{2}-2z=1$$

Combine 4z and -2z to get 2z.

$$2z+9-z^{2}=1$$

Subtract 1 from both sides.

$$2z+9-z^{2}-1=0$$

Subtract 1 from 9 to get 8.

$$2z+8-z^{2}=0$$

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

$$-z^{2}+2z+8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -z^{2}+az+bz+8. To find a and b, set up a system to be solved.

$$a+b=2 ab=-8=-8$$

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 -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 2.

$$a=4 b=-2$$

Rewrite -z^{2}+2z+8 as \left(-z^{2}+4z\right)+\left(-2z+8\right).

$$\left(-z^{2}+4z\right)+\left(-2z+8\right)$$

Factor out -z in the first and -2 in the second group.

$$-z\left(z-4\right)-2\left(z-4\right)$$

Factor out common term z-4 by using distributive property.

$$\left(z-4\right)\left(-z-2\right)$$

To find equation solutions, solve z-4=0 and -z-2=0.

$$z=4 z=-2$$

Substitute 4 for z in the equation \sqrt{4z+9}=z+1.

$$\sqrt{4\times 4+9}=4+1$$

Simplify. The value z=4 satisfies the equation.

$$5=5$$

Substitute -2 for z in the equation \sqrt{4z+9}=z+1.

$$\sqrt{4\left(-2\right)+9}=-2+1$$

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

$$1=-1$$

Equation \sqrt{4z+9}=z+1 has a unique solution.

$$z=4$$