What is the quadratic formula?

It solves ax^2 + bx + c = 0 using x = (-b +/- sqrt(b^2 - 4ac)) / (2a). It works for any quadratic with a not equal to 0.

What is the discriminant and what does it tell us?

The discriminant is b^2 - 4ac. If it is positive, there are two real roots; if zero, one repeated real root; if negative, two complex roots.

How do I identify a, b, and c in a quadratic equation?

In ax^2 + bx + c = 0: a is the coefficient of x^2, b is the coefficient of x, c is the constant term. Example: 2x^2 - 5x + 3 = 0 has a = 2, b = -5, c = 3. Make sure the equation equals zero first.

What are complex roots?

Complex roots occur when the discriminant is negative. They appear as conjugate pairs a + bi and a - bi, where i = sqrt(-1).

What is the vertex form of a quadratic?

Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex. From standard form, h = -b/(2a) and k = f(h).

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Solve quadratic equations ax²+bx+c=0 using the quadratic formula with step-by-step solutions

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Enter coefficients for the equation ax² + bx + c = 0

Coefficient a (x² term)

Coefficient b (x term)

Coefficient c (constant term)

Try These Examples

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Example 1

x² - 5x + 6 = 0 (basic case)

a: 1

b: -5

c: 6

Example 2

2x² + 3x - 2 = 0 (medium difficulty)

a: 2

b: 3

c: -2

Example 3

x² - 4x + 4 = 0 (perfect square)

a: 1

b: -4

c: 4

Example 4

x² + 2x + 5 = 0 (complex roots)

a: 1

b: 2

c: 5

Example 5

-3x² + 12x - 9 = 0 (negative coefficient)

a: -3

b: 12

c: -9

Example 6

0.5x² - 2x + 1.5 = 0 (decimal coefficients)

a: 0.5

b: -2

c: 1.5

Quadratic Formula

The quadratic formula solves equations of the form ax² + bx + c = 0:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression under the square root (b² - 4ac) is called the discriminant.

Discriminant Analysis

Δ > 0: Two distinct real roots

Δ = 0: One repeated real root (perfect square)

Δ < 0: Two complex conjugate roots

\Delta = b^2 - 4ac

Real-World Applications

Physics: Projectile Motion

Calculate when a thrown object hits the ground using h = -16t² + v₀t + h₀, where h is height, t is time, and v₀ is initial velocity.

Business: Profit Optimization

Find optimal pricing to maximize revenue using profit functions like P = -2x² + 100x - 500, where x represents price points.

Engineering: Bridge Design

Model parabolic arches and determine structural dimensions for suspension bridges and architectural curved surfaces.

Agriculture: Area Optimization

Calculate dimensions for fencing rectangular fields with fixed perimeter to maximize planting area.

Computer Graphics: Curve Generation

Generate smooth parabolic curves for animations, game trajectories, and bezier curve approximations.

Common Mistakes to Avoid

❌

Forgetting the ± symbol

The quadratic formula has ±, which means you must calculate both + and - to get both roots.

❌

Sign errors with negative b

The formula uses -b, so if b is already negative, -(-5) becomes +5. Watch your signs carefully.

❌

Incorrect order of operations

Calculate the entire numerator (-b ± √Δ) before dividing by 2a. Don't divide -b by 2a first.

❌

Using formula when a = 0

If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula doesn't apply.

✅

Best Practice

Always calculate the discriminant first to determine the nature of roots before proceeding with the full formula.

Quadratic Equation Solution Methods Comparison

Method	When to Use	Advantages	Disadvantages
Factoring	Simple equations with integer roots	Fast and intuitive; builds algebraic understanding	Only works for factorable equations; difficult with complex coefficients
Quadratic Formula	Any quadratic equation	Works for ALL quadratics; systematic approach; handles complex roots	More steps; requires careful calculation
Completing the Square	Deriving vertex form; theoretical work	Provides vertex directly; deepens conceptual understanding	More algebraic manipulation; prone to arithmetic errors
Graphing	Visual understanding; approximate solutions	Shows parabola shape; finds vertex and intercepts visually	Often gives approximate values; time-consuming

Understanding Parabola Properties

Every quadratic equation ax² + bx + c = 0 corresponds to a parabola when graphed. Understanding parabola properties helps visualize solutions.

Vertex (Turning Point)

h = -\frac{b}{2a}, \quad k = f(h)

The vertex represents the maximum or minimum value of the quadratic function.

Axis of Symmetry

x = -\frac{b}{2a}

The parabola is symmetric about this vertical line passing through the vertex.

Direction of Opening

If a > 0: Opens upward (U-shaped)
If a < 0: Opens downward (∩-shaped)

Roots and x-intercepts

The solutions from the quadratic formula are the x-coordinates where the parabola crosses the x-axis.

Connection to Discriminant: The discriminant tells you how many times the parabola crosses the x-axis: Δ > 0 (two crossings), Δ = 0 (touches once at vertex), Δ < 0 (doesn't cross).

How NASA Engineers Use Quadratics to Launch Rockets

Every object thrown, launched, or dropped follows a parabolic path described by a quadratic equation. NASA uses this exact math to calculate when rockets reach maximum altitude and when debris returns to Earth.

Real Problem: Model Rocket Launch

A rocket is launched with initial velocity 174 ft/s from a platform 78 ft high. The height equation is:

h(t) = -16t^2 + 174t + 78

Here, $-16$ comes from half of Earth's gravitational acceleration ( $\frac{1}{2}g = \frac{1}{2} \times 32 \text{ ft/s}^2$ ).

🕐 When does it hit the ground?

Set h(t) = 0 and solve:

-16t^2 + 174t + 78 = 0

\Delta = 174^2 - 4(-16)(78) = 30276 + 4992 = 35268

t = \frac{-174 \pm \sqrt{35268}}{2(-16)} = \frac{-174 \pm 187.8}{-32}

t ≈ 11.3 seconds (the positive root — the negative root is physically meaningless)

📈 Maximum altitude?

The vertex of the parabola gives the peak:

t_{\text{peak}} = -\frac{b}{2a} = -\frac{174}{2(-16)} = 5.44 \text{ s}

h_{\text{max}} = -16(5.44)^2 + 174(5.44) + 78

h_max ≈ 551 feet (about 168 meters)

Discriminant connection: Since Δ = 35,268 > 0, the parabola crosses the x-axis twice — meaning the rocket goes up and comes back down. If Δ were negative, the rocket would never reach ground level (physically impossible in this context, but possible if the equation modeled something with an offset).

📖 NASA Space Math 📖 Khan Academy: Projectile Motion

Frequently Asked Questions

The quadratic formula x = (-b ± √(b² - 4ac)) / 2a is the universal solution for finding the roots of any quadratic equation in the form ax² + bx + c = 0. It was derived by completing the square on the general quadratic equation.