📐 Enter Coefficients of ax² + bx + c = 0

1x² + 0x + 0 = 0

a (x² coefficient)

b (x coefficient)

c (constant)

Try:

—

Discriminant (Δ = b²−4ac)

—

Solutions

📊 Parabola Properties

Equation—

Parabola Opens—

Vertex (h, k)—

Axis of Symmetry—

Y-Intercept—

X-Intercept(s)—

Sum of Roots—

Product of Roots—

Discriminant (Δ)—

Nature of Roots—

📝 Step-by-Step Solution

🔲 Completing the Square

📈 Parabola Graph

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Parabola Roots (x-intercepts) Vertex Y-intercept

📐 Quadratic Formula & Methods

The Quadratic Formula

For any equation ax² + bx + c = 0 (where a ≠ 0):

x = (−b ± √(b² − 4ac)) / (2a)

The ± gives TWO solutions:
x₁ = (−b + √(b²−4ac)) / (2a)
x₂ = (−b − √(b²−4ac)) / (2a)

The Discriminant (Δ = b² − 4ac)

Δ > 0 → Two distinct real roots (parabola crosses x-axis twice)
Δ = 0 → One repeated real root (parabola touches x-axis once)
Δ < 0 → Two complex conjugate roots (parabola doesn't cross x-axis)

Completing the Square

ax² + bx + c = 0
Step 1: x² + (b/a)x + c/a = 0
Step 2: x² + (b/a)x = -c/a
Step 3: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
Step 4: (x + b/2a)² = (b²-4ac)/(4a²)
Step 5: x = -b/2a ± √(b²-4ac)/(2a)

Factoring Method

Find two numbers m and n such that m×n = ac and m+n = b, then factor.

x² − 5x + 6 = 0
Find m×n = 6, m+n = −5: → m=−2, n=−3
(x − 2)(x − 3) = 0
x = 2 or x = 3

Vertex Form

Standard: y = ax² + bx + c
Vertex: y = a(x − h)² + k

Where vertex = (h, k):
h = −b / (2a)
k = c − b²/(4a) OR f(h)

Vieta's Formulas

Relations between coefficients and roots (x₁, x₂):

Sum of roots: x₁ + x₂ = −b/a
Product of roots: x₁ × x₂ = c/a

Example: x²−5x+6=0 → roots 2,3
Sum: 2+3 = 5 = −(−5)/1 ✓
Product: 2×3 = 6 = 6/1 ✓

❓ Frequently Asked Questions

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