Solve y-9sqrt{y}+14=0

$$y - 9 \sqrt { y } + 14 = 0$$

Solve for y

$$y=49$$

$$y=4$$

Solution Steps

Subtract 14 from both sides. Anything subtracted from zero gives its negation.

$$y-9\sqrt{y}=-14$$

Subtract y from both sides of the equation.

$$-9\sqrt{y}=-14-y$$

Square both sides of the equation.

$$\left(-9\sqrt{y}\right)^{2}=\left(-14-y\right)^{2}$$

Expand \left(-9\sqrt{y}\right)^{2}.

$$\left(-9\right)^{2}\left(\sqrt{y}\right)^{2}=\left(-14-y\right)^{2}$$

Calculate -9 to the power of 2 and get 81.

$$81\left(\sqrt{y}\right)^{2}=\left(-14-y\right)^{2}$$

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

$$81y=\left(-14-y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-14-y\right)^{2}.

$$81y=196+28y+y^{2}$$

Subtract 28y from both sides.

$$81y-28y=196+y^{2}$$

Combine 81y and -28y to get 53y.

$$53y=196+y^{2}$$

Subtract y^{2} from both sides.

$$53y-y^{2}=196$$

Subtract 196 from both sides.

$$53y-y^{2}-196=0$$

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

$$-y^{2}+53y-196=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -y^{2}+ay+by-196. To find a and b, set up a system to be solved.

$$a+b=53 ab=-\left(-196\right)=196$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 196.

$$1,196 2,98 4,49 7,28 14,14$$

Calculate the sum for each pair.

$$1+196=197 2+98=100 4+49=53 7+28=35 14+14=28$$

The solution is the pair that gives sum 53.

$$a=49 b=4$$

Rewrite -y^{2}+53y-196 as \left(-y^{2}+49y\right)+\left(4y-196\right).

$$\left(-y^{2}+49y\right)+\left(4y-196\right)$$

Factor out -y in the first and 4 in the second group.

$$-y\left(y-49\right)+4\left(y-49\right)$$

Factor out common term y-49 by using distributive property.

$$\left(y-49\right)\left(-y+4\right)$$

To find equation solutions, solve y-49=0 and -y+4=0.

$$y=49 y=4$$

Substitute 49 for y in the equation y-9\sqrt{y}+14=0.

$$49-9\sqrt{49}+14=0$$

Simplify. The value y=49 satisfies the equation.

$$0=0$$

Substitute 4 for y in the equation y-9\sqrt{y}+14=0.

$$4-9\sqrt{4}+14=0$$

Simplify. The value y=4 satisfies the equation.

$$0=0$$

List all solutions of -9\sqrt{y}=-y-14.

$$y=49 y=4$$