Solve sqrt{10y+24}-3=y+1

$$\sqrt { 10 y + 24 } - 3 = y + 1$$

Solve for y

$$y=4$$

$$y=-2$$

Solution Steps

Subtract -3 from both sides of the equation.

$$\sqrt{10y+24}=y+1+3$$

Add 1 and 3 to get 4.

$$\sqrt{10y+24}=y+4$$

Square both sides of the equation.

$$\left(\sqrt{10y+24}\right)^{2}=\left(y+4\right)^{2}$$

Calculate \sqrt{10y+24} to the power of 2 and get 10y+24.

$$10y+24=\left(y+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+4\right)^{2}.

$$10y+24=y^{2}+8y+16$$

Subtract y^{2} from both sides.

$$10y+24-y^{2}=8y+16$$

Subtract 8y from both sides.

$$10y+24-y^{2}-8y=16$$

Combine 10y and -8y to get 2y.

$$2y+24-y^{2}=16$$

Subtract 16 from both sides.

$$2y+24-y^{2}-16=0$$

Subtract 16 from 24 to get 8.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -y^{2}+ay+by+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 -y^{2}+2y+8 as \left(-y^{2}+4y\right)+\left(-2y+8\right).

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

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

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

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

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

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

$$y=4 y=-2$$

Substitute 4 for y in the equation \sqrt{10y+24}-3=y+1.

$$\sqrt{10\times 4+24}-3=4+1$$

Simplify. The value y=4 satisfies the equation.

$$5=5$$

Substitute -2 for y in the equation \sqrt{10y+24}-3=y+1.

$$\sqrt{10\left(-2\right)+24}-3=-2+1$$

Simplify. The value y=-2 satisfies the equation.

$$-1=-1$$

List all solutions of \sqrt{10y+24}=y+4.

$$y=4 y=-2$$