Answer:
Case 1: 2 solutions
Case 2: 1 solution
Case 3: none/imaginary solutions
Step-by-step explanation:
The discriminant comes from the square root section of the quadratic formula:
[tex]x= \frac{-b±\sqrt{b^{2}-4ac}}{2a}[/tex] (ignore the Â, it's a formatting error)
The quadratic formula is used to find the solutions for quadratic equations. Quadratic equations are in the format ax² + bx + c. To solve for "x", you would substitute "a", "b" and "c" into the formula.
Why the discriminant works:
[tex]\sqrt{b^{2}-4ac }[/tex]
When you have to square root a number, it has to be positive inside the root (or else you get imaginary numbers).
Example: (if the inside of the root is less than 0)
If b = 4, a = 5, c = 1 :
b² - 4ac
= 4² - 4(5)(1)
= 16 - 20
= -4
√-4 is imaginary. The quadratic equation has no solutions because you can't do the rest of the quadratic formula.
Example: (inside of the root is 0)
If b = 4, a = 4, c = 1 :
b² - 4ac
= 4² - 4(4)(1)
= 16 - 16
= 0
There will be one solution. In the quadratic formula, this affects where there is a ± sign. See the formula again.
[tex]x= \frac{-b±\sqrt{b^{2}-4ac}}{2a}[/tex] If the square root area equals 0:
[tex]x= \frac{-b±0}{2a}[/tex] Adding and subtracting 0 give the same answer.
Therefore there is only one solution.
Example: (inside of the root is greater than 0)
If b = 4, a = 3, c = 1 :
b² - 4ac
= 4² - 4(3)(1)
= 16 - 12
= 4
There will be two solutions. Notice where the ± sign is again in the uadratic formula.
[tex]x= \frac{-b±\sqrt{b^{2}-4ac}}{2a}[/tex]
[tex]x= \frac{-b±\sqrt{4}}{2a}[/tex] Substitute the answer for b² - 4ac
[tex]x= \frac{-b±2}{2a}[/tex] You can get two different answers between adding and subtracting 2.
