Quadratic Formula Calculator - Solve Equations Online

Free quadratic formula calculator and solver. Master the quadratic formula with step-by-step solutions, quadratic formula practice problems, and interactive learning. What is the quadratic formula? Learn the quadratic formula through examples and quadratic formula derivation.

🧮 Quadratic Formula Calculator📚 Quadratic Formula Solutions🎯 Quadratic Formula Practice📊 Quadratic Formula Graphs

Quadratic Formula Calculator

Enter the coefficients for the equation $ax^2 + bx + c = 0$.

🚀How to Use the Quadratic Formula Calculator - Step by Step Guide

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Enter Coefficients

Input the coefficients a, b, and c from your quadratic equation to use the quadratic formula solver

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Calculate Solutions

Click "Calculate" to see the quadratic formula calculator apply the quadratic formula with detailed solutions

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Study the Process

Learn from the detailed quadratic formula derivation, quadratic formula discriminant analysis, and mathematical concepts

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Practice & Visualize

Use quadratic formula practice mode and explore the parabola graph to deepen your quadratic formula understanding

💡 Learning Strategy

Start with simple quadratic formula examples like x² - 5x + 6 = 0. Understanding the quadratic formula discriminant (b² - 4ac) is key to predicting solution types before applying the quadratic formula.

📐What is the Quadratic Formula? Complete Understanding

🔢Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree 2, containing one unknown variable (usually x). The standard form is:

ax^2 + bx + c = 0

where a, b, and c are known constants, and $a \neq 0$ .

🎯What is the Quadratic Formula Used For?

The quadratic formula provides a universal method to find the roots (solutions) of any quadratic equation. This quadratic formula calculator applies the quadratic formula automatically:

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

🔍The Discriminant and Solution Types

In the quadratic formula, the discriminant $\Delta = b^2 - 4ac$ determines the nature of solutions:

✅ $\Delta > 0$ : Two distinct real roots
⚪ $\Delta = 0$ : One repeated real root
❌ $\Delta < 0$ : Two complex conjugate roots

📈Geometric Interpretation

Every quadratic equation corresponds to a parabola $y = ax^2 + bx + c$ . The roots represent where the parabola intersects the x-axis, and the vertex occurs at $x = -\frac{b}{2a}$ .

🧮Quadratic Formula Derivation - Complete Mathematical Proof

🎯Goal: Derive the Quadratic Formula from First Principles

We will derive the quadratic formula by completing the square on the general quadratic equation. This quadratic formula derivation method transforms the equation into a form where we can easily solve for x using the quadratic formula.

\text{Starting equation: } ax^2 + bx + c = 0 \text{ where } a \neq 0

Step 1: Isolate the constant term

Move the constant term c to the right side of the equation.

ax^2 + bx = -c

Step 2: Divide by coefficient a

Divide the entire equation by a to make the coefficient of x² equal to 1.

x^2 + \frac{b}{a}x = -\frac{c}{a}

Step 3: Complete the square

Add and subtract $\left(\frac{b}{2a}\right)^2$ to complete the square on the left side.

x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2

Step 4: Factor the perfect square

The left side is now a perfect square trinomial that can be factored.

\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}

Step 5: Simplify the right side

Combine fractions on the right side using a common denominator.

\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}

Step 6: Take the square root

Take the square root of both sides, remembering the ± symbol.

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

Step 7: Solve for x

Subtract $\frac{b}{2a}$ from both sides to isolate x.

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

Step 8: Final quadratic formula

Combine the fractions to get the final quadratic formula that our quadratic formula calculator uses.

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

🔗Understanding the Discriminant (b² - 4ac)

The expression under the square root, $b^2 - 4ac$ , is called the discriminant and is crucial for understanding the nature of solutions:

b^2 - 4ac > 0

Two distinct real roots

b^2 - 4ac = 0

One repeated real root

b^2 - 4ac < 0

Two complex conjugate roots

📊Vieta's Formulas: Relationship Between Roots and Coefficients

If $x_1$ and $x_2$ are the roots of $ax^2 + bx + c = 0$ , then:

Sum of Roots

x_1 + x_2 = -\frac{b}{a}

Product of Roots

x_1 \cdot x_2 = \frac{c}{a}

These relationships, known as Vieta's formulas, provide valuable insights into quadratic equations without directly solving them.

Interactive Practice Problems

Click "Generate New Problem" to start practicing

Enter the two solutions of this equation (x₁ and x₂):

Generate Printable Quadratic Formula Worksheet

Click the button below to generate a PDF worksheet with 12 quadratic formula practice problems for offline learning.

Include answer key

Quadratic Formula Practice Worksheet

Name: _______________ Date: _______________

Answer Key

❓Quadratic Formula FAQ - Common Questions Answered

What is the quadratic formula and when do I use it?

The quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ is used to solve any quadratic equation ax² + bx + c = 0. Use the quadratic formula when factoring is difficult or impossible, or when you need exact decimal solutions. Our quadratic formula calculator applies the quadratic formula automatically.

How do I solve quadratic equations step by step?

1) Identify coefficients a, b, c. 2) Calculate the discriminant b² - 4ac. 3) Apply the quadratic formula. 4) Simplify the solutions. Our calculator shows each step in detail.

What are quadratic formula examples for beginners?

Try simple quadratic formula examples: x² - 5x + 6 = 0 (factors nicely), x² + 2x - 1 = 0 (requires quadratic formula), or x² + x + 1 = 0 (complex solutions). Our quadratic formula practice mode provides guided quadratic formula examples with step-by-step solutions.

How is the quadratic formula derived?

The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. Start by dividing by a, then add and subtract (b/2a)² to create a perfect square trinomial.

What does the discriminant tell us about solutions?

The discriminant b² - 4ac predicts solution types: positive gives two real roots, zero gives one repeated root, negative gives complex roots. It's like a preview before solving.

Who invented the quadratic formula?

The quadratic formula evolved over centuries. Ancient Babylonians (2000 BCE) had geometric methods, Al-Khwarizmi (9th century) systematized algebraic approaches, and modern notation developed during the Renaissance.

Are there quadratic formula practice worksheets available?

Yes! Our quadratic formula practice mode generates unlimited quadratic formula problems with instant feedback. You can also create printable quadratic formula worksheets with 12 problems and answer keys for offline quadratic formula practice.

How do I remember the quadratic formula?

Try the famous quadratic formula song: "x equals negative b, plus or minus the square root, of b squared minus four a c, all over two a". Rhythm helps memory!

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