Solve y^2-5y+6

$$y ^ { 2 } - 5 y + 6$$

Factor

$$\left(y-3\right)\left(y-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+6. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\times 6=6$$

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 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -5.

$$a=-3 b=-2$$

Rewrite y^{2}-5y+6 as \left(y^{2}-3y\right)+\left(-2y+6\right).

$$\left(y^{2}-3y\right)+\left(-2y+6\right)$$

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

$$y\left(y-3\right)-2\left(y-3\right)$$

Factor out common term y-3 by using distributive property.

$$\left(y-3\right)\left(y-2\right)$$

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.

$$y^{2}-5y+6=0$$

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.

$$y=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6}}{2}$$

Square -5.

$$y=\frac{-\left(-5\right)±\sqrt{25-4\times 6}}{2}$$

Multiply -4 times 6.

$$y=\frac{-\left(-5\right)±\sqrt{25-24}}{2}$$

Add 25 to -24.

$$y=\frac{-\left(-5\right)±\sqrt{1}}{2}$$

Take the square root of 1.

$$y=\frac{-\left(-5\right)±1}{2}$$

The opposite of -5 is 5.

$$y=\frac{5±1}{2}$$

Now solve the equation y=\frac{5±1}{2} when ± is plus. Add 5 to 1.

$$y=\frac{6}{2}$$

Divide 6 by 2.

$$y=3$$

Now solve the equation y=\frac{5±1}{2} when ± is minus. Subtract 1 from 5.

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

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 3 for x_{1} and 2 for x_{2}.

$$y^{2}-5y+6=\left(y-3\right)\left(y-2\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -5x +6 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 5 rs = 6$$

Two numbers r and s sum up to 5 exactly when the average of the two numbers is \frac{1}{2}*5 = \frac{5}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{5}{2} - u s = \frac{5}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 6

$$(\frac{5}{2} - u) (\frac{5}{2} + u) = 6$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{25}{4} - u^2 = 6$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = 6-\frac{25}{4} = -\frac{1}{4}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{5}{2} - \frac{1}{2} = 2 s = \frac{5}{2} + \frac{1}{2} = 3$$

Evaluate

$$\left(y-3\right)\left(y-2\right)$$