Quadratic Formula Calculator
Solve quadratic equations (ax^2 + bx + c = 0) step-by-step using the quadratic formula, compute discriminant (Δ = b^2 - 4ac), find real/complex roots, vertex (h, k), and axis of symmetry.
ax² + bx + c = 0 Quadratic Solver
Enter coefficients a, b, and constant c.
Quadratic Equation Output
Quick Summary
Our quadratic formula calculator discriminant steps tool solves second-degree polynomial equations $ax^2 + bx + c = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. It evaluates the discriminant ($\Delta = b^2 - 4ac$), handles real and complex conjugate roots ($\alpha \pm \beta i$), and computes the parabola vertex $(h, k)$.
How It Works: Discriminant & Parabola Geometry
A quadratic equation forms a U-shaped parabolic curve on a graph:
1. **Quadratic Formula:** $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
2. **Discriminant ($\Delta = b^2 - 4ac$):**
• If $\Delta > 0$: Two distinct real roots.
• If $\Delta = 0$: One repeated real root ($x = -\b / 2a$).
• If $\Delta < 0$: Two complex conjugate roots ($\alpha \pm \beta i$).
3. **Parabola Vertex ($h, k$):** Peak or trough at $h = -\b / 2a, k = c - \b^2 / 4a$.
4. **Axis of Symmetry:** Vertical line passing through vertex at $x = -\b / 2a$.
Formula Explanation
Your quadratic calculations follow classical algebraic quadratic equations:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for solving x² - 5x + 6 = 0:
- Step 1 (Identify Coefficients): $a = 1, b = -5, c = 6$.
- Step 2 (Calculate Discriminant Δ): $\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = \mathbf{1} > 0$ (Two real roots).
- Step 3 (Apply Quadratic Formula): $x = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \5 \pm 1 / 2$.
- Step 4 (Evaluate Roots): $x_1 = \5 + 1 / 2 = \6 / 2 = \mathbf{3}$, $x_2 = \5 - 1 / 2 = \4 / 2 = \mathbf{2}$.
- Step 5 (Verify Solutions): $3^2 - 5(3) + 6 = 9 - 15 + 6 = 0$ and $2^2 - 5(2) + 6 = 4 - 10 + 6 = 0$, delivering **x₁ = 3, x2 = 2**!
Calculation Examples: Real-World Scenario Comparison
Compare discriminants, root types, and calculated solutions across quadratic equations:
| Quadratic Equation | Coefficients (a, b, c) | Discriminant (Δ) | Root Status | Calculated Roots | Vertex (h, k) |
|---|---|---|---|---|---|
| x² - 5x + 6 = 0 | (1, -5, 6) | 1 | Two Real Roots | x&sb1; = 3, x&sb2; = 2 | (2.5, -0.25) |
| x² - 4x + 4 = 0 | (1, -4, 4) | 0 | One Double Root | x = 2.00 | (2.0, 0.0) |
| x² + 2x + 5 = 0 | (1, 2, 5) | -16 | Two Complex Roots | -1 ± 2i | (-1.0, 4.0) |
| 2x² - 4x - 6 = 0 | (2, -4, -6) | 64 | Two Real Roots | x&sb1; = 3, x&sb2; = -1 | (1.0, -8.0) |
Benefits of Using the Quadratic Formula Calculator
Utilizing this calculator provides essential physics projectile motion, business profit maximization, and algebra test advantages:
- Handles Complex Conjugate Roots: Calculates imaginary roots ($\alpha \pm \beta i$) using the imaginary unit $i = \sqrt{-1}$ when $\Delta < 0$.
- Discriminant Analysis: Instantly indicates whether an equation has 2 real roots, 1 double root, or 2 complex roots.
- Physics Projectile Motion ($h(t) = -16t^2 + v_0 t + h_0$): Calculates time $t$ when a launched object hits the ground ($h = 0$).
- Business Profit Maximization: Finds the revenue or profit peak vertex $(h, k)$ of quadratic cost functions.
Frequently Asked Questions (FAQ)
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to solve any equation of the form ax² + bx + c = 0.
What is the discriminant (Δ)?
The discriminant is the expression under the square root: Δ = b² - 4ac. It determines the number and type of roots.
What does Δ > 0 mean?
Δ > 0 means the quadratic equation has two distinct real roots (the parabola crosses the x-axis twice).
What does Δ = 0 mean?
Δ = 0 means the equation has one repeated real root (the vertex touches the x-axis exactly once).
What does Δ < 0 mean?
Δ < 0 means the equation has two complex conjugate roots involving imaginary unit i (α ± βi) (the parabola does not touch the x-axis).
How do you find the parabola vertex (h, k)?
x-coordinate h = -b / (2a), and y-coordinate k = c - b² / (4a).
How do you solve x² - 5x + 6 = 0?
a=1, b=-5, c=6. Δ = (-5)² - 4(1)(6) = 1. x = (5 ± 1) / 2 ⇒ x1 = 3, x2 = 2.
What is the axis of symmetry of a parabola?
The vertical line x = -b / (2a) that divides the parabola symmetrically into two mirror halves.
Can coefficient 'a' be zero in a quadratic equation?
NO! If a = 0, the x² term disappears, making it a linear equation (bx + c = 0), not a quadratic equation.
Why is the quadratic formula useful when factoring fails?
Factoring only works for nice integer roots, but the quadratic formula works for ALL quadratic equations including irrational decimals and complex numbers.