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4.6 The Quadratic Formula and the Discriminant
Objectives:1. Solve quadratic equations by using
the Quadratic Formula2. Use the discriminant to determine
the number and type of roots for a quadratic equation.
Quadratic Formula
Always works to solve a quadratic equation, but is a little lengthy.
The solutions of a quadratic equation of the form where a≠0 are given by the formula:
2 4
2
b b acx
a
2 0ax bx c
Example
x²-8x=33x²-8x-33=0 Set = 0a=1, b=-8, c=-33
2 4
2
b b acx
a
28 ( 8) 4(1)( 33)
2(1)x
8 64 132
2x
8 196
2x
8 14
2x
8 14
2x
8 14
2x
22
2x 6
2x
11x 3x { 3,11}
Another Example7x²+6x+2=0a=7, b=6, c=2
Since ALL of the coefficients are divisible by 2, simplify by dividing them by 2.
26 (6) 4(7)(2)
2(7)x
6 36 56
14x
6 20
14x
6 2 5
14
ix
3 5
7
ix
Discriminant
The discriminant describes the solution to a quadratic equation. The part of the quadratic formula under the radical is the discriminant or b²-4ac.
• If b2 – 4ac > 0, and a perfect square– You have two rational roots
• If b2 – 4ac >0, and not a perfect square.– You have two irrational roots
• If b2 – 4ac = 0– You have 1 real, rational root. (Repeated root)
• If b2 – 4ac < 0– You have two complex roots
ExamplesFind the discriminant and
describe the number and type of roots.
a. x²-16x+64=0b²-4ac
(-16)²-4(1)(64)=256-256=0
One real, rational root because the discriminant equals 0.
b. 7x²-3x=0(-3)²-4(7)(0)=9-0=9
Two rational roots because 9 is positive and a perfect square.
c. 3x²-x+5=0(-1)²-4(3)(5)=1-60=-59
Two complex roots because the discriminant is a negative.
We have discussed several methods for solving quadratic equations – which one do you use?
Method Can Be Used When to Use
Graphing sometimes Use only if an exact number is not required. Best use to check the reasonableness of solutions found algebraically
Factoring sometimes Use if the constant term is 0 or if the factors are easily determined
Square Root Property sometimes Use for equations in which a perfect square is equal to a constant
Completing the Square always Useful for equations of the form ax2 + bx + c where b is even
Quadratic Formula always When other methods fail or are too tedious
Solve – use any method.1. 7x²-14x=07x(x-2)=0x=0, x=22. x²-64=0
x²=64x=8
3. x²-16x+64=0(x-8)(x-8)=0x=8
4. x²+5x+8=0Doesn’t factor, not easily
done by completing the square (5 is odd) so use quadratic formula.
25 5 4(1)(8)
2(1)x
5 25 32
2x
5 7
2x
5 7
2
ix
HomeworkWorkbook Page 55
