🔣 🔣 Quadratic Equation Solver: The Quadratic Formula Explained

A quadratic equation is any equation in the form ax² + bx + c = 0, where a ≠ 0. Quadratics appear in physics (projectile motion), engineering (beam deflection), finance (break-even analysis), and geometry (area problems). Three methods solve them: factoring, completing the square, and the quadratic formula — the universal method that always works.

Standard Form of a Quadratic Equation

ax² + bx + c = 0

Where a, b, and c are constants and a ≠ 0. Examples:

• x² − 5x + 6 = 0 (a=1, b=−5, c=6)
• 2x² + 3x − 2 = 0 (a=2, b=3, c=−2)
• x² − 9 = 0 (a=1, b=0, c=−9)

Method 1: The Quadratic Formula (Always Works)

x = (−b ± √(b² − 4ac)) ÷ (2a)

The ± means there are typically two solutions (roots).

Example: Solve x² − 5x + 6 = 0

a=1, b=−5, c=6

• x = (−(−5) ± √((−5)² − 4×1×6)) ÷ (2×1)
• x = (5 ± √(25 − 24)) ÷ 2
• x = (5 ± √1) ÷ 2
• x = (5 ± 1) ÷ 2
• x = 6/2 = 3 or x = 4/2 = 2

Example: Solve 2x² + 3x − 2 = 0

a=2, b=3, c=−2

• x = (−3 ± √(9 − 4×2×(−2))) ÷ (2×2)
• x = (−3 ± √(9 + 16)) ÷ 4
• x = (−3 ± √25) ÷ 4
• x = (−3 ± 5) ÷ 4
• x = 2/4 = 0.5 or x = −8/4 = −2

The Discriminant: Knowing How Many Solutions

The expression inside the square root, b² − 4ac, is called the discriminant (Δ). Its sign tells you the nature of the roots before you fully solve:

• Δ > 0: Two distinct real roots (the parabola crosses the x-axis at two points)
• Δ = 0: One repeated real root (the parabola just touches the x-axis)
• Δ < 0: No real roots — two complex conjugate roots (the parabola never crosses the x-axis)

Examples of Each Case

• x² − 5x + 6 = 0: Δ = 25 − 24 = 1 > 0 → two real roots (3 and 2)
• x² − 4x + 4 = 0: Δ = 16 − 16 = 0 → one repeated root (x = 2)
• x² + x + 1 = 0: Δ = 1 − 4 = −3 < 0 → no real roots (complex: x = (−1 ± i√3)/2)

Method 2: Factoring (When It Works Cleanly)

If the quadratic factors as (x − r₁)(x − r₂) = 0, the roots are r₁ and r₂.

x² − 5x + 6 = 0: Need two numbers that multiply to +6 and add to −5 → −2 and −3

(x − 2)(x − 3) = 0 → x = 2 or x = 3 ✓

Factoring is fastest when it works, but doesn't always produce integers. When in doubt, use the quadratic formula.

Method 3: Completing the Square

Useful for understanding the structure and deriving the quadratic formula itself.

Solve x² + 6x + 5 = 0:

• Move constant: x² + 6x = −5
• Add (6/2)² = 9 to both sides: x² + 6x + 9 = 4
• Factor left side: (x + 3)² = 4
• Take square root: x + 3 = ±2
• Solve: x = −1 or x = −5

Real-World Quadratic Applications

Projectile Motion

A ball thrown upward with initial velocity 20 m/s from height 5 m:

Height h(t) = −4.9t² + 20t + 5

When does it hit the ground? Set h = 0: −4.9t² + 20t + 5 = 0

Using quadratic formula: t ≈ 4.32 seconds (ignoring the negative solution)

Break-Even Analysis

Revenue and cost curves often produce quadratic intersection problems. If revenue = −x² + 100x and cost = 500 + 20x, break-even when revenue = cost:

−x² + 100x = 500 + 20x → −x² + 80x − 500 = 0 → x² − 80x + 500 = 0

Using quadratic formula: x ≈ 6.7 or 73.3 units

Area Problems

A rectangle has perimeter 28 m and area 40 m². Find its dimensions.

If length = l and width = w: l + w = 14 and l × w = 40

Substituting: l(14 − l) = 40 → 14l − l² = 40 → l² − 14l + 40 = 0

Quadratic formula: l = (14 ± √(196−160))/2 = (14 ± 6)/2 → l = 10 or l = 4

Dimensions: 10 m × 4 m