Deriving and Using the Quadratic Formula: A Step-by-Step Guide
Solve any quadratic equation instantly with step-by-step working using our Quadratic Formula Calculator, or read our comprehensive guide below.
In algebra, a quadratic equation is any equation that can be rearranged in standard form as:
ax² + bx + c = 0
Where x represents an unknown, and a, b, and c represent known numbers, where a is not equal to zero. When you graph a quadratic equation, it forms a U-shaped curve called a parabola. Solving the equation means finding where the parabola crosses the x-axis. These points are called the "roots" or "x-intercepts."
The Quadratic Formula
While some quadratic equations can be factored easily, the quadratic formula is a universal method that works for all quadratic equations, no matter how complex the numbers are.
x = [-b ± √(b² - 4ac)] / 2a
Step-by-Step Example
Let's solve the equation: 2x² - 5x - 3 = 0
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Identify a, b, and c:
- a = 2
- b = -5
- c = -3
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Plug into the formula: x = [-(-5) ± √((-5)² - 4 * 2 * -3)] / (2 * 2)
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Simplify the discriminant (the part under the square root): Discriminant = 25 - (-24) = 49
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Calculate the square root: √49 = 7
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Solve for the two possible values of x (using ±):
- Option 1 (Plus): x = (5 + 7) / 4 = 12 / 4 = 3
- Option 2 (Minus): x = (5 - 7) / 4 = -2 / 4 = -0.5
The roots of the equation are 3 and -0.5.
Understanding the Discriminant
The expression underneath the square root, b² - 4ac, is called the discriminant. It tells you exactly how many real solutions the equation has before you even finish the math:
| Discriminant Value | Number of Roots | Graph Appearance | |--------------------|-----------------|------------------| | Positive (>0) | Two real roots | The parabola crosses the x-axis in two places. | | Zero (=0) | One real root | The tip (vertex) of the parabola touches the x-axis exactly once. | | Negative (<0) | No real roots (Two complex/imaginary roots) | The parabola does not touch the x-axis at all. |
FAQ
Why do we need the ± symbol? Because a square root has both a positive and negative solution. For example, 3² is 9, but (-3)² is also 9. The ± ensures we find both places where the parabola crosses the x-axis.
Can 'a' ever be zero? No. If 'a' is zero, the x² term disappears. The equation becomes bx + c = 0, which is a linear equation (a straight line), not a quadratic equation (a parabola).
For complex equations that require immediate answers with step-by-step breakdown, use the CalcUnit Quadratic Formula Calculator.