Solve sqrt{y}+2y=105

$$\sqrt{ y } +2y=105$$

Solve for y

$$y=49$$

Solution Steps

Subtract 2y from both sides of the equation.

$$\sqrt{y}=105-2y$$

Square both sides of the equation.

$$\left(\sqrt{y}\right)^{2}=\left(105-2y\right)^{2}$$

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

$$y=\left(105-2y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(105-2y\right)^{2}.

$$y=11025-420y+4y^{2}$$

Subtract 11025 from both sides.

$$y-11025=-420y+4y^{2}$$

Add 420y to both sides.

$$y-11025+420y=4y^{2}$$

Combine y and 420y to get 421y.

$$421y-11025=4y^{2}$$

Subtract 4y^{2} from both sides.

$$421y-11025-4y^{2}=0$$

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.

$$-4y^{2}+421y-11025=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 421 for b, and -11025 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-421±\sqrt{421^{2}-4\left(-4\right)\left(-11025\right)}}{2\left(-4\right)}$$

Square 421.

$$y=\frac{-421±\sqrt{177241-4\left(-4\right)\left(-11025\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-421±\sqrt{177241+16\left(-11025\right)}}{2\left(-4\right)}$$

Multiply 16 times -11025.

$$y=\frac{-421±\sqrt{177241-176400}}{2\left(-4\right)}$$

Add 177241 to -176400.

$$y=\frac{-421±\sqrt{841}}{2\left(-4\right)}$$

Take the square root of 841.

$$y=\frac{-421±29}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-421±29}{-8}$$

Now solve the equation y=\frac{-421±29}{-8} when ± is plus. Add -421 to 29.

$$y=-\frac{392}{-8}$$

Divide -392 by -8.

$$y=49$$

Now solve the equation y=\frac{-421±29}{-8} when ± is minus. Subtract 29 from -421.

$$y=-\frac{450}{-8}$$

Reduce the fraction \frac{-450}{-8} to lowest terms by extracting and canceling out 2.

$$y=\frac{225}{4}$$

The equation is now solved.

$$y=49 y=\frac{225}{4}$$

Substitute 49 for y in the equation \sqrt{y}+2y=105.

$$\sqrt{49}+2\times 49=105$$

Simplify. The value y=49 satisfies the equation.

$$105=105$$

Substitute \frac{225}{4} for y in the equation \sqrt{y}+2y=105.

$$\sqrt{\frac{225}{4}}+2\times \frac{225}{4}=105$$

Simplify. The value y=\frac{225}{4} does not satisfy the equation.

$$120=105$$

Equation \sqrt{y}=105-2y has a unique solution.

$$y=49$$