01 Key Concepts
The Quadratic Formula
For ax^2+bx+c=0, x = [-b plus or minus sqrt(b^2-4ac)] / (2a).
Identifying a, b, c
Match your equation to standard form ax^2+bx+c=0 and read off the three coefficients directly.
The Discriminant
The expression b^2-4ac inside the square root. It determines the nature of the solutions without fully solving the equation.
Interpreting the Discriminant
If b^2-4ac > 0: two distinct real solutions. If = 0: exactly one repeated real solution. If < 0: no real solutions (two complex solutions).
When to Use the Quadratic Formula
Use it whenever factoring is difficult or impossible — it always works for any quadratic equation.
02 Key Formulas
- x = [-b plus or minus sqrt(b^2 - 4ac)] / (2a)
- Discriminant = b^2 - 4ac
03 Solved Examples
Example 1 Solve x^2 + 5x + 6 = 0 using the quadratic formula.
- Identify a=1, b=5, c=6.
- Discriminant: b^2-4ac = 25-24 = 1.
- x = [-5 plus or minus sqrt(1)] / 2 = (-5 plus or minus 1)/2.
- Two solutions: (-5+1)/2 = -2, and (-5-1)/2 = -3.
Answer: x = -2 or x = -3
Example 2 Find the discriminant of 2x^2 + 3x + 5 = 0 and describe the solutions.
- Identify a=2, b=3, c=5.
- Discriminant = b^2-4ac = 9 - 40.
Answer: Discriminant = -31 (negative), so there are no real solutions
Example 3 Solve x^2 - 4x + 4 = 0 using the quadratic formula.
- Identify a=1, b=-4, c=4.
- Discriminant: (-4)^2-4(1)(4) = 16-16 = 0.
- x = [4 plus or minus sqrt(0)] / 2 = 4/2.
Answer: x = 2 (a repeated root, since discriminant = 0)
04 Practice Questions
1Find a, b, c for 3x^2 - 2x + 7 = 0.
a=3, b=-2, c=7
2Find the discriminant of x^2 + 2x + 1 = 0.
0
3Find the discriminant of x^2 + x + 1 = 0.
-3
4Solve x^2 - 3x - 4 = 0 using the quadratic formula.
x=4 or x=-1
5If the discriminant is positive, how many real solutions exist?
Two distinct real solutions
📄 Quadratic Formula — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.