Solve y^2-9y+14

$${ y }^{ 2 } -9y+14$$

Factor

$$\left(y-7\right)\left(y-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+14. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\times 14=14$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 14.

$$-1,-14 -2,-7$$

Calculate the sum for each pair.

$$-1-14=-15 -2-7=-9$$

The solution is the pair that gives sum -9.

$$a=-7 b=-2$$

Rewrite y^{2}-9y+14 as \left(y^{2}-7y\right)+\left(-2y+14\right).

$$\left(y^{2}-7y\right)+\left(-2y+14\right)$$

Factor out y in the first and -2 in the second group.

$$y\left(y-7\right)-2\left(y-7\right)$$

Factor out common term y-7 by using distributive property.

$$\left(y-7\right)\left(y-2\right)$$

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.

$$y^{2}-9y+14=0$$

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.

$$y=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 14}}{2}$$

Square -9.

$$y=\frac{-\left(-9\right)±\sqrt{81-4\times 14}}{2}$$

Multiply -4 times 14.

$$y=\frac{-\left(-9\right)±\sqrt{81-56}}{2}$$

Add 81 to -56.

$$y=\frac{-\left(-9\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$y=\frac{-\left(-9\right)±5}{2}$$

The opposite of -9 is 9.

$$y=\frac{9±5}{2}$$

Now solve the equation y=\frac{9±5}{2} when ± is plus. Add 9 to 5.

$$y=\frac{14}{2}$$

Divide 14 by 2.

$$y=7$$

Now solve the equation y=\frac{9±5}{2} when ± is minus. Subtract 5 from 9.

$$y=\frac{4}{2}$$

Divide 4 by 2.

$$y=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 7 for x_{1} and 2 for x_{2}.

$$y^{2}-9y+14=\left(y-7\right)\left(y-2\right)$$

Evaluate

$$\left(y-7\right)\left(y-2\right)$$