QUADRATIC EQUATIONS

DEFINITION:

For a, b, c R and a 0 z , the expressions in the form of 2 ax bx c 0 are called quadratic
equations with variable x where the real numbers a, b, c are called coefficients

The values of x (they can be real or nonreal) which satisfy the given equation are called the roats of the equation and the set containing the roots is called solution set.

Example:

If 3 2b 4 (a 1)x x (a b)x a.b 0 is a quadratic equation, find a and b.
a-1=0
a=1
2b-4=2
2b=2+4
2b=6
b=3

Discriminant

Do you see b2 − 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:

when b2 − 4ac is positive, we get two Real solutions

when it is zero we get just ONE real solution (both answers are the same)

when it is negative we get a pair of Complex solutions

Using the Quadratic Formula

exapmles:

Coefficients are: a = 5, b = 6, c = 1

Quadratic Formula: x = −b ± √(b2 − 4ac)/2a

Put in a, b and c: x = −6 ± √(62 − 4×5×1)/2×5

Solve: x = −6 ± √(36 − 20)/10

x = −6 ± √(16)/10

x = −6 ± 4/10

Answer: x = −0,2 or x = −1

x = −0,2 or −1

Check -0,2:

5×(−0,2)2 + 6×(−0,2) + 1

= 5×(0,04) + 6×(−0,2) + 1

= 0,2 − 1,2 + 1

= 0

Check -1:

5×(−1)2 + 6×(−1) + 1

= 5×(1) + 6×(−1) + 1

= 5 − 6 + 1

= 0

Example:

Solve x2 − 4x + 6,25 = 0

click to edit

Note that the Discriminant is negative:
b2 − 4ac = (−4)2 − 4×1×6,25=-9

Use the Quadratic Formula: x = −(−4) ± √(−9)/2

√(−9) = 3i
(where i is the imaginary number √−1)

So: x = 4 ± 3i/2

Answer: x = 2 ± 1,5i

The graph does not cross the x-axis. That is why we ended up with complex numbers.

BUT an upside-down mirror image of our equation does cross the x-axis at 2 ± 1,5 (note: missing the i).