Discover the powerful quadratic formula that can solve any quadratic equation. Learn how the discriminant reveals the nature of solutions before you even calculate them.
For any quadratic equation: ax² + bx + c = 0 (where a ≠ 0)
a: coefficient of x² (quadratic coefficient)
b: coefficient of x (linear coefficient)
c: constant term
±: indicates two possible solutions
√(b² - 4ac): the discriminant (determines solution type)
Find a, b, and c from ax² + bx + c = 0
Find b² - 4ac to determine solution type
Replace a, b, c in the quadratic formula
Calculate both solutions using + and -
Solve: 2x² - 7x + 3 = 0
Step 1: Identify coefficients
a = 2, b = -7, c = 3
Step 2: Calculate discriminant
b² - 4ac = (-7)² - 4(2)(3) = 49 - 24 = 25
Since 25 > 0, there are two real solutions
Step 3: Apply quadratic formula
Step 4: Calculate both solutions
x = (7 + 5)/4
x = 3
x = (7 - 5)/4
x = 1/2
Solutions: x = 3 or x = 1/2
The discriminant tells us about the nature of solutions without solving the equation!
Two real, distinct solutions
x = (-b + √D)/(2a)
x = (-b - √D)/(2a)
One real solution (double root)
x = -b/(2a)
No real solutions (complex)
x = (-b ± i√|D|)/(2a)
Solve: x² - 6x + 9 = 0
Step 1: Identify coefficients
a = 1, b = -6, c = 9
Step 2: Calculate discriminant
b² - 4ac = (-6)² - 4(1)(9) = 36 - 36 = 0
Since 0 = 0, there is one real solution
Step 3: Apply quadratic formula
Solution: x = 3 (double root)
Solve: x² + 4x + 5 = 0
Step 1: Identify coefficients
a = 1, b = 4, c = 5
Step 2: Calculate discriminant
b² - 4ac = (4)² - 4(1)(5) = 16 - 20 = -4
Since -4 < 0, there are no real solutions
Step 3: Complex solutions (for reference)
No real solutions (complex: x = -2 ± i)
In x² - 5x + 6 = 0, b = -5 (not 5). The formula uses -b, so -(-5) = +5
Always divide the entire numerator by 2a, not just the discriminant part
Always simplify √(b² - 4ac) when possible (e.g., √25 = 5, not √25)
Problem 1:
Solve: x² - 8x + 12 = 0
Discriminant: 64 - 48 = 16 > 0 (two real solutions)
x = (8 ± 4)/2
Solutions: x = 6 or x = 2
Problem 2:
Solve: 4x² - 4x + 1 = 0
Discriminant: 16 - 16 = 0 (one real solution)
x = 4/8 = 1/2
Solution: x = 1/2 (double root)
Problem 3:
Solve: x² + 2x + 3 = 0
Discriminant: 4 - 12 = -8 < 0 (no real solutions)
Complex solutions: x = -1 ± i√2
Maximum/Minimum Values
Use vertex formula: x = -b/(2a)
Example: Find maximum area of rectangle with perimeter 20
When Δ < 0
Solutions involve √(-1) = i
Example: x² + 4x + 5 = 0 has complex solutions
Bridge Design
Finding optimal arch height
Calculating stress points and load distribution
Investment Analysis
Finding break-even points
Risk assessment and portfolio optimization
Algorithm Design
Time complexity analysis
Graphics programming and 3D modeling