The Quadratic Formula
Lets complete the square for
ax2 + bx + c
Now what if we want to solve the equation
ax2 + bx + c = 0
We can equivalently solve
by the root method
b2 - 4ac
Now take a square root of both sides to get
Finally subtract b/2a from both sides to get the quadratic formula.
Memorize This Formula!
3x2 - 2x + 5
a = 3 b = -2 c = 5
We define the discriminant as
D = b2 - 4ac
is a convenient measure of determining how many real roots (solutions) there are. Notice that D is the expression inside of the square root sign in the quadratic formula. Since the square root of a negative number produces only complex numbers, we see that if D is negative, then there will be no real roots. If D is a positive number, then the quadratic formula will produce two roots (one for the plus and one for the minus). If D is 0 then plus 0 and minus 0 are the same number, so we get only one root. The table below summarizes.
How many roots are there for the equation:
3x2 - 5x + 1 = 0
D = 25 - 12 > 0
hence there are two real roots.
The Sum and Product of the Roots
Since the two roots of a quadratic are
then if we add the two roots, we get:
and if we multiply the two roots we get
Note that if a is 1 then the sum of the roots is -b and the product is c. This relates to factoring when we find two numbers that add to b and multiply to c.
What are the sum and product of the roots of
4x2 -3x + 2
The sum is
and the product of the roots is
Determining the Quadratic Equation From the Roots
If we know the roots of a quadratic then it is easy to find the original quadratic by using the zero product formula in reverse.
Find an equation of a quadratic that has roots 2 and -4/3.
We can write:
(x - 2)(x - (-4/3)) = (x - 2)(x + 4/3)
Find a quadratic with the following roots