Solve sqrt{5n+9}=n-1

$$\sqrt { 5 n + 9 } = n - 1$$

Solve for n

$$n=8$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{5n+9}\right)^{2}=\left(n-1\right)^{2}$$

Calculate \sqrt{5n+9} to the power of 2 and get 5n+9.

$$5n+9=\left(n-1\right)^{2}$$

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

$$5n+9=n^{2}-2n+1$$

Subtract n^{2} from both sides.

$$5n+9-n^{2}=-2n+1$$

Add 2n to both sides.

$$5n+9-n^{2}+2n=1$$

Combine 5n and 2n to get 7n.

$$7n+9-n^{2}=1$$

Subtract 1 from both sides.

$$7n+9-n^{2}-1=0$$

Subtract 1 from 9 to get 8.

$$7n+8-n^{2}=0$$

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

$$-n^{2}+7n+8=0$$

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

$$a+b=7 ab=-8=-8$$

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 7.

$$a=8 b=-1$$

Rewrite -n^{2}+7n+8 as \left(-n^{2}+8n\right)+\left(-n+8\right).

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

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

$$-n\left(n-8\right)-\left(n-8\right)$$

Factor out common term n-8 by using distributive property.

$$\left(n-8\right)\left(-n-1\right)$$

To find equation solutions, solve n-8=0 and -n-1=0.

$$n=8 n=-1$$

Substitute 8 for n in the equation \sqrt{5n+9}=n-1.

$$\sqrt{5\times 8+9}=8-1$$

Simplify. The value n=8 satisfies the equation.

$$7=7$$

Substitute -1 for n in the equation \sqrt{5n+9}=n-1.

$$\sqrt{5\left(-1\right)+9}=-1-1$$

Simplify. The value n=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

Equation \sqrt{5n+9}=n-1 has a unique solution.

$$n=8$$