In Algebra 1, the discriminant is used to verify that there are no real solutions or roots for quadratic equations.
When solving a quadratic equation, we are usually searching for the two solutions that will make the equation true. We have encounter a few situations where there was only one solution, or even no real number solution to the equation.
Is there a way to "know" whether there will be 0, 1 or 2 solutions to a quadratic equation?
The answer is "yes", and is connected to the quadratic formula.
The discriminant is the portion "under" the square root symbol
in the Quadratic Formula: |
The value under the radical (square root) portion of the quadratic formula indicates the number of solutions for that specific quadratic equation. That value is "b2 - 4ac".
b2 - 4ac is called the discriminant. |
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| Note: The discriminant, b2 - 4ac, does NOT tell you what the solution(s) to the quadratic equation will be. It simply tells you "how many" real solutions there will be.
If the discriminant is a positive number = 2 real roots.
If the discriminant is zero = 1 real root.
If the discriminant is a negative number = 0 real roots. |
DISCRIMINANT: Its purpose is to tell "how many roots", and "what type of roots". |
POSITIVE
b2 - 4ac > 0 |
ZERO
b2 - 4ac = 0 |
NEGATIVE
b2 - 4ac < 0 |
x2 + 6 x + 5 = 0
discriminant:
b2 - 4ac = 62 - 4(1)(5)
= 16 (positive)
There are two real roots.
There are two x-intercepts.
If the discriminant is a perfect square, the two roots are rational numbers. If the discriminant is not a perfect square, the two roots are irrational numbers containing a radical.
x2 + 6x + 5 = (x + 1)(x + 5) = 0
Roots: x = -1, x = -5 |
x2 - 2x+ 1 = 0
discriminant:
b2 - 4ac = (-2)2-4(1)(1)
= 0 (zero)
There is one real root.
There is one x-intercept.
(The root is repeated.)
x2 - 2x+ 1 = (x - 1)2
= (x - 1)(x - 1) = 0
Repeated root: x = 1
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x2 - 3x + 10 = 0
discriminant:
b2 - 4ac = (-3)2-4(1)(10)
= -31 (negative)
There are NO real roots.
There are no x-intercepts.
In this situation, there will be two "complex" roots because there will be a negative number under the square root. The answers involve the imaginary number, i.
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