Quadratic Formula Calculator

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2, expressed in the standard form:

ax² + bx + c = 0

Where:

• a, b, and c are constants (coefficients)
• a ≠ 0 (if a = 0, it becomes a linear equation)
• x is the variable (unknown value we're solving for)

Quadratic equations appear frequently in mathematics, physics, engineering, and economics—from calculating projectile motion to optimizing business profits.

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The Quadratic Formula

The quadratic formula provides a systematic method to find the solutions (roots) of any quadratic equation:

x = (-b ± √(b² - 4ac)) / (2a)

This formula always works for any quadratic equation, regardless of whether the roots are real numbers or complex numbers.

When to Use the Quadratic Formula

While there are multiple methods to solve quadratic equations (factoring, completing the square, graphing), the quadratic formula is particularly useful when:

• The equation cannot be easily factored
• You need precise decimal answers
• You're working with coefficients that are fractions or decimals
• The equation has complex (imaginary) roots

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Understanding the Discriminant

The discriminant is the expression under the square root in the quadratic formula:

Δ = b² - 4ac

The discriminant tells us about the nature of the roots before we even calculate them:

Discriminant Value	Type of Roots	Example
Δ > 0	Two distinct real roots	x² - 5x + 6 = 0 (Δ = 1)
Δ = 0	One repeated real root	x² + 6x + 9 = 0 (Δ = 0)
Δ < 0	Two complex conjugate roots	x² + 2x + 5 = 0 (Δ = -16)

Why the Discriminant Matters

• Δ > 0: The parabola crosses the x-axis at two points
• Δ = 0: The parabola touches the x-axis at exactly one point (the vertex)
• Δ < 0: The parabola doesn't cross the x-axis at all (roots are imaginary)

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Step-by-Step Examples

Example 1: Two Distinct Real Roots (Δ > 0)

Problem: Solve x² - 5x + 6 = 0

Solution:

1. Identify coefficients: a = 1, b = -5, c = 6
2. Calculate discriminant:
   • Δ = b² - 4ac
   • Δ = (-5)² - 4(1)(6)
   • Δ = 25 - 24 = 1 ✓ (positive, so two real roots)
3. Apply quadratic formula:
   • x = (5 ± √1) / 2
   • x = (5 ± 1) / 2
4. Calculate both roots:
   • x₁ = (5 + 1) / 2 = 3
   • x₂ = (5 - 1) / 2 = 2

Answer: x = 2 or x = 3

Verification: (2)² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓

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Example 2: One Repeated Root (Δ = 0)

Problem: Solve x² + 6x + 9 = 0

Solution:

1. Identify coefficients: a = 1, b = 6, c = 9
2. Calculate discriminant:
   • Δ = 6² - 4(1)(9)
   • Δ = 36 - 36 = 0 ✓ (zero, so one repeated root)
3. Apply formula:
   • x = -6 / 2(1)
   • x = -3

Answer: x = -3 (with multiplicity 2)

Note: This equation can also be factored as (x + 3)² = 0

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Example 3: Complex Roots (Δ < 0)

Problem: Solve x² + 2x + 5 = 0

Solution:

1. Identify coefficients: a = 1, b = 2, c = 5
2. Calculate discriminant:
   • Δ = 2² - 4(1)(5)
   • Δ = 4 - 20 = -16 ✓ (negative, so complex roots)
3. Apply formula with imaginary numbers:
   • x = (-2 ± √(-16)) / 2
   • x = (-2 ± 4i) / 2
4. Simplify:
   • x₁ = -1 + 2i
   • x₂ = -1 - 2i

Answer: x = -1 ± 2i (complex conjugates)

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Common Mistakes to Avoid

1. Forgetting the ± Symbol

❌ Wrong: x = (-b + √Δ) / 2a (only one root)
✓ Correct: x = (-b ± √Δ) / 2a (both roots)

2. Order of Operations Errors

❌ Wrong: x = -b ± √Δ / 2a (dividing only the square root)
✓ Correct: x = (-b ± √Δ) / (2a) (dividing the entire numerator)

3. Sign Errors with Negative b

When b is negative (e.g., b = -5), -b becomes positive:
-b = -(-5) = +5

4. Forgetting to Divide by 2a

The entire numerator (-b ± √Δ) must be divided by 2a, not just a.

5. Complex Number Notation

❌ Wrong: x = -1 + 4i / 2 = -1 + 2i²
✓ Correct: x = (-1 + 4i) / 2 = -0.5 + 2i

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Applications of Quadratic Equations

1. Physics: Projectile Motion

When you throw a ball, its height h at time t follows:
h = -4.9t² + v₀t + h₀

Use the quadratic formula to find when the ball hits the ground (h = 0).

2. Business: Profit Maximization

Profit P often relates to price p quadratically:
P = -ap² + bp + c

The quadratic formula helps find the optimal price for maximum profit.

3. Engineering: Structural Analysis

Designing arches, bridges, and parabolic structures requires solving quadratic equations.

4. Computer Graphics

Rendering curves, calculating intersections, and animation paths use quadratic equations extensively.

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Frequently Asked Questions

What if the coefficient a = 0?

If a = 0, the equation becomes bx + c = 0, which is a linear equation, not quadratic. It has only one solution: x = -c/b. The quadratic formula requires a ≠ 0.

Can I always use the quadratic formula?

Yes! The quadratic formula works for every quadratic equation. However, factoring may be faster for simple equations like x² - 4 = 0, which factors to (x-2)(x+2) = 0.

How do I check if my answer is correct?

Substitute your solutions back into the original equation:

• If ax² + bx + c = 0 after substitution, your answer is correct
• Example: For x² - 5x + 6 = 0 with x = 2:
  • (2)² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓

What are complex/imaginary roots?

Complex roots occur when the discriminant is negative (Δ < 0). They involve the imaginary unit i, where i² = -1. Complex roots always come in conjugate pairs: a + bi and a - bi.

Quadratic Formula vs Factoring: Which is Better?

Factoring is faster when:

• The equation is simple (e.g., x² - 9 = 0)
• Coefficients are small integers

Quadratic Formula is better when:

• Factoring is difficult or impossible
• You need decimal precision
• Coefficients are fractions/decimals
• Roots are irrational or complex

How do I simplify radical expressions in the answer?

If the discriminant is a perfect square, simplify:

• √4 = 2
• √9 = 3
• √16 = 4

If not a perfect square, leave in radical form or use decimal approximation:

• √5 ≈ 2.236
• √13 ≈ 3.606

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References

This calculator is based on fundamental algebraic principles and verified against authoritative mathematics resources:

• Khan Academy: Quadratic Formula
• Wolfram MathWorld: Quadratic Equation
• Paul's Online Math Notes: Quadratic Equations
• Math is Fun: Quadratic Equations