First, some general instruction for you:
You can ALWAYS factor a quadratic with the quadratic formula. A quadratic is any expression or equation where the highest power of x is 2, that is, a second degree polynomial. The quadratic equation will give you two values for x, which are the x-intercepts of the parabola. Sometimes when using the quadratic equation, you will end up taking the square root of a negative number, introducing "I" and showing that the x-intercepts are imaginary... the parabola does not cross the x-axis.
For your reference: the quadratic equation is...
x = [ -b ± √(b^2 - 4ac) ] / 2a
The plus/minus sign ± is what gives you the two answers instead of just one.
After using the quadratic equation to find the two values for x, some students have trouble converting the x-values into the factors. Just solve the x = n answers to get zero on one side. The other side will be your parenthetical factor. If n is a fraction, set the equation equal to zero using two steps. First multiply both sides by the denominator, then add or subtract the numerator from both sides.
For example, if one answer from the quadratic formula is x = 3/4, then to find the factor, multiply both sides by 4...
4x = 3
Then subtract three from both sides...
4x - 3 = 0.
The factor would be (4x - 3)
Now to the problem at hand:
For this one, there is no need to use the quadratic formula. Any time you want to factor, the first step is to see if you can pull out a common factor from each term.
In this problem, there is a common factor of 2.
2x^2-22x+56=0
2(x^2-11x+28)=0
Now factor the parenthesis by considering which two factors of +28 will give you -11. Those factors are -7 and -4.
So the equation factors to...
2(x-7)(x-4)=0
This is your final answer.
(Review "factoring of trinomials" if need be.)
If you want to go a bit further, realize that the two factors show you the x-intecepts fo the graph of the function
y = 2x^2-22x+56
The x-intercepts (or "roots") of the parabola (quadratic) would be 4 and 7.
The axis of symmetry for the parabola would be half way between 4 and 7 at x = 5.5.
To find the vertex, plug in the x-value of 6.5 to find the y-value of the vertex.
f(5.5) = 2 (6.5)^2 - 22(6.5) + 56
f(5.5) = -2.25
So the vertex would be (5.5, -2.25).
In addition, it is easy to find the y-intercept. Simply find f(0). This easily figures to 56.
That gives you all you need to sketch the parabola.
