Solve y^2-28y+165=0

$${ y }^{ 2 } -28y+165 = 0$$

Solve for y

$$y=\sqrt{31}+14\approx 19.567764363$$

$$y=14-\sqrt{31}\approx 8.432235637$$

Steps Using the Quadratic Formula

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.

$$y^{2}-28y+165=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -28 for b, and 165 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 165}}{2}$$

Square -28.

$$y=\frac{-\left(-28\right)±\sqrt{784-4\times 165}}{2}$$

Multiply -4 times 165.

$$y=\frac{-\left(-28\right)±\sqrt{784-660}}{2}$$

Add 784 to -660.

$$y=\frac{-\left(-28\right)±\sqrt{124}}{2}$$

Take the square root of 124.

$$y=\frac{-\left(-28\right)±2\sqrt{31}}{2}$$

The opposite of -28 is 28.

$$y=\frac{28±2\sqrt{31}}{2}$$

Now solve the equation y=\frac{28±2\sqrt{31}}{2} when ± is plus. Add 28 to 2\sqrt{31}.

$$y=\frac{2\sqrt{31}+28}{2}$$

Divide 28+2\sqrt{31} by 2.

$$y=\sqrt{31}+14$$

Now solve the equation y=\frac{28±2\sqrt{31}}{2} when ± is minus. Subtract 2\sqrt{31} from 28.

$$y=\frac{28-2\sqrt{31}}{2}$$

Divide 28-2\sqrt{31} by 2.

$$y=14-\sqrt{31}$$

The equation is now solved.

$$y=\sqrt{31}+14 y=14-\sqrt{31}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$y^{2}-28y+165=0$$

Subtract 165 from both sides of the equation.

$$y^{2}-28y+165-165=-165$$

Subtracting 165 from itself leaves 0.

$$y^{2}-28y=-165$$

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-28y+\left(-14\right)^{2}=-165+\left(-14\right)^{2}$$

Square -14.

$$y^{2}-28y+196=-165+196$$

Add -165 to 196.

$$y^{2}-28y+196=31$$

Factor y^{2}-28y+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-14\right)^{2}=31$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-14\right)^{2}}=\sqrt{31}$$

Simplify.

$$y-14=\sqrt{31} y-14=-\sqrt{31}$$

Add 14 to both sides of the equation.

$$y=\sqrt{31}+14 y=14-\sqrt{31}$$