Solve 3(sqrt{5-10y})+y+11=0

$$3 ( \sqrt { 5 - 10 y } ) + y + 11 = 0$$

Solve for y

$$y=-6\sqrt{85}-56\approx -111.317266744$$

Solution Steps

Subtract y+11 from both sides of the equation.

$$3\sqrt{5-10y}=-\left(y+11\right)$$

To find the opposite of y+11, find the opposite of each term.

$$3\sqrt{5-10y}=-y-11$$

Square both sides of the equation.

$$\left(3\sqrt{5-10y}\right)^{2}=\left(-y-11\right)^{2}$$

Expand \left(3\sqrt{5-10y}\right)^{2}.

$$3^{2}\left(\sqrt{5-10y}\right)^{2}=\left(-y-11\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{5-10y}\right)^{2}=\left(-y-11\right)^{2}$$

Calculate \sqrt{5-10y} to the power of 2 and get 5-10y.

$$9\left(5-10y\right)=\left(-y-11\right)^{2}$$

Use the distributive property to multiply 9 by 5-10y.

$$45-90y=\left(-y-11\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-y-11\right)^{2}.

$$45-90y=y^{2}+22y+121$$

Subtract y^{2} from both sides.

$$45-90y-y^{2}=22y+121$$

Subtract 22y from both sides.

$$45-90y-y^{2}-22y=121$$

Combine -90y and -22y to get -112y.

$$45-112y-y^{2}=121$$

Subtract 121 from both sides.

$$45-112y-y^{2}-121=0$$

Subtract 121 from 45 to get -76.

$$-76-112y-y^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-y^{2}-112y-76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -112 for b, and -76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\left(-1\right)\left(-76\right)}}{2\left(-1\right)}$$

Square -112.

$$y=\frac{-\left(-112\right)±\sqrt{12544-4\left(-1\right)\left(-76\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-\left(-112\right)±\sqrt{12544+4\left(-76\right)}}{2\left(-1\right)}$$

Multiply 4 times -76.

$$y=\frac{-\left(-112\right)±\sqrt{12544-304}}{2\left(-1\right)}$$

Add 12544 to -304.

$$y=\frac{-\left(-112\right)±\sqrt{12240}}{2\left(-1\right)}$$

Take the square root of 12240.

$$y=\frac{-\left(-112\right)±12\sqrt{85}}{2\left(-1\right)}$$

The opposite of -112 is 112.

$$y=\frac{112±12\sqrt{85}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{112±12\sqrt{85}}{-2}$$

Now solve the equation y=\frac{112±12\sqrt{85}}{-2} when ± is plus. Add 112 to 12\sqrt{85}.

$$y=\frac{12\sqrt{85}+112}{-2}$$

Divide 112+12\sqrt{85} by -2.

$$y=-6\sqrt{85}-56$$

Now solve the equation y=\frac{112±12\sqrt{85}}{-2} when ± is minus. Subtract 12\sqrt{85} from 112.

$$y=\frac{112-12\sqrt{85}}{-2}$$

Divide 112-12\sqrt{85} by -2.

$$y=6\sqrt{85}-56$$

The equation is now solved.

$$y=-6\sqrt{85}-56 y=6\sqrt{85}-56$$

Substitute -6\sqrt{85}-56 for y in the equation 3\sqrt{5-10y}+y+11=0.

$$3\sqrt{5-10\left(-6\sqrt{85}-56\right)}-6\sqrt{85}-56+11=0$$

Simplify. The value y=-6\sqrt{85}-56 satisfies the equation.

$$0=0$$

Substitute 6\sqrt{85}-56 for y in the equation 3\sqrt{5-10y}+y+11=0.

$$3\sqrt{5-10\left(6\sqrt{85}-56\right)}+6\sqrt{85}-56+11=0$$

Simplify. The value y=6\sqrt{85}-56 does not satisfy the equation.

$$12\times 85^{\frac{1}{2}}-90=0$$

Equation 3\sqrt{5-10y}=-y-11 has a unique solution.

$$y=-6\sqrt{85}-56$$