How to Solve Quadratic Equations - Quadratic Formula Guide

You're staring at 2x² − 7x + 3 = 0 and factoring isn't working. The quadratic formula is your universal escape hatch — it solves every quadratic equation, no matter how ugly the numbers.

Here's how to use it, step by step.

The Quadratic Formula

For any equation ax² + bx + c = 0 (where a ≠ 0):

x = (−b ± √(b² − 4ac)) / 2a

Three steps: identify a, b, c → plug them in → simplify.

Step-by-Step Examples

Example 1: Two Real Roots

Solve 2x² − 7x + 3 = 0

1. a = 2, b = −7, c = 3
2. Discriminant = b² − 4ac = 49 − 24 = 25
3. x = (7 ± √25) / 4 = (7 ± 5) / 4
4. x₁ = (7+5)/4 = 3 and x₂ = (7−5)/4 = 0.5

✅ Two real solutions: x = 3 and x = 0.5

Example 2: One Repeated Root

Solve x² − 6x + 9 = 0

1. a = 1, b = −6, c = 9
2. Discriminant = 36 − 36 = 0
3. x = (6 ± √0) / 2 = 6/2 = 3

✅ One repeated solution: x = 3 (the parabola just touches the x-axis)

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

1. a = 1, b = 2, c = 5
2. Discriminant = 4 − 20 = −16
3. x = (−2 ± √(−16)) / 2 = (−2 ± 4i) / 2
4. x₁ = −1 + 2i and x₂ = −1 − 2i

✅ Two complex solutions — the parabola never crosses the x-axis.

The Discriminant — Your Crystal Ball

The discriminant Δ = b² − 4ac tells you what kind of solutions you'll get before you finish the calculation:

| Discriminant | Solutions | Graph Behavior | |-------------|-----------|---------------| | Δ > 0 | Two distinct real roots | Crosses x-axis twice | | Δ = 0 | One repeated real root | Touches x-axis once | | Δ < 0 | Two complex roots | Never touches x-axis |

The Trench Truth: Always compute the discriminant first. If it's negative, you know immediately the answer involves imaginary numbers — no point trying to factor. If it's a perfect square, the roots are rational and you might have been able to factor. The discriminant saves you from wasted effort.

Finding the Vertex and Axis of Symmetry

Every parabola has a vertex — its peak or valley:

Vertex x-coordinate: x = −b/(2a)

For 2x² − 7x + 3: x = 7/4 = 1.75

Plug back in: y = 2(1.75)² − 7(1.75) + 3 = 6.125 − 12.25 + 3 = −3.125

Vertex: (1.75, −3.125) — this is the minimum since a > 0.

Try our quadratic formula calculator to solve any equation instantly with full step-by-step solutions.

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