Solve y^2-32y+44

$${ y }^{ 2 } -32y+44$$

Factor

$$\left(y-\left(16-2\sqrt{53}\right)\right)\left(y-\left(2\sqrt{53}+16\right)\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$y^{2}-32y+44=0$$

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.

$$y=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 44}}{2}$$

Square -32.

$$y=\frac{-\left(-32\right)±\sqrt{1024-4\times 44}}{2}$$

Multiply -4 times 44.

$$y=\frac{-\left(-32\right)±\sqrt{1024-176}}{2}$$

Add 1024 to -176.

$$y=\frac{-\left(-32\right)±\sqrt{848}}{2}$$

Take the square root of 848.

$$y=\frac{-\left(-32\right)±4\sqrt{53}}{2}$$

The opposite of -32 is 32.

$$y=\frac{32±4\sqrt{53}}{2}$$

Now solve the equation y=\frac{32±4\sqrt{53}}{2} when ± is plus. Add 32 to 4\sqrt{53}.

$$y=\frac{4\sqrt{53}+32}{2}$$

Divide 32+4\sqrt{53} by 2.

$$y=2\sqrt{53}+16$$

Now solve the equation y=\frac{32±4\sqrt{53}}{2} when ± is minus. Subtract 4\sqrt{53} from 32.

$$y=\frac{32-4\sqrt{53}}{2}$$

Divide 32-4\sqrt{53} by 2.

$$y=16-2\sqrt{53}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 16+2\sqrt{53} for x_{1} and 16-2\sqrt{53} for x_{2}.

$$y^{2}-32y+44=\left(y-\left(2\sqrt{53}+16\right)\right)\left(y-\left(16-2\sqrt{53}\right)\right)$$

Evaluate

$$y^{2}-32y+44$$