Solve sqrt{y}+2y=153

$$\sqrt { y } + 2 y = 153$$

Solve for y

$$y = \frac{289}{4} = 72\frac{1}{4} = 72.25$$

Solution Steps

Subtract 2y from both sides of the equation.

$$\sqrt{y}=153-2y$$

Square both sides of the equation.

$$\left(\sqrt{y}\right)^{2}=\left(153-2y\right)^{2}$$

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

$$y=\left(153-2y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(153-2y\right)^{2}.

$$y=23409-612y+4y^{2}$$

Subtract 23409 from both sides.

$$y-23409=-612y+4y^{2}$$

Add 612y to both sides.

$$y-23409+612y=4y^{2}$$

Combine y and 612y to get 613y.

$$613y-23409=4y^{2}$$

Subtract 4y^{2} from both sides.

$$613y-23409-4y^{2}=0$$

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.

$$-4y^{2}+613y-23409=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 613 for b, and -23409 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-613±\sqrt{613^{2}-4\left(-4\right)\left(-23409\right)}}{2\left(-4\right)}$$

Square 613.

$$y=\frac{-613±\sqrt{375769-4\left(-4\right)\left(-23409\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-613±\sqrt{375769+16\left(-23409\right)}}{2\left(-4\right)}$$

Multiply 16 times -23409.

$$y=\frac{-613±\sqrt{375769-374544}}{2\left(-4\right)}$$

Add 375769 to -374544.

$$y=\frac{-613±\sqrt{1225}}{2\left(-4\right)}$$

Take the square root of 1225.

$$y=\frac{-613±35}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-613±35}{-8}$$

Now solve the equation y=\frac{-613±35}{-8} when ± is plus. Add -613 to 35.

$$y=-\frac{578}{-8}$$

Reduce the fraction \frac{-578}{-8} to lowest terms by extracting and canceling out 2.

$$y=\frac{289}{4}$$

Now solve the equation y=\frac{-613±35}{-8} when ± is minus. Subtract 35 from -613.

$$y=-\frac{648}{-8}$$

Divide -648 by -8.

$$y=81$$

The equation is now solved.

$$y=\frac{289}{4} y=81$$

Substitute \frac{289}{4} for y in the equation \sqrt{y}+2y=153.

$$\sqrt{\frac{289}{4}}+2\times \frac{289}{4}=153$$

Simplify. The value y=\frac{289}{4} satisfies the equation.

$$153=153$$

Substitute 81 for y in the equation \sqrt{y}+2y=153.

$$\sqrt{81}+2\times 81=153$$

Simplify. The value y=81 does not satisfy the equation.

$$171=153$$

Equation \sqrt{y}=153-2y has a unique solution.

$$y=\frac{289}{4}$$