Quadratic Equation Examples With Step-by-Step Solutions
Follow quadratic examples with two real roots, one repeated root, complex roots, a non-monic equation, discriminants, vertices, and graph interpretation.
How should you use these Quadratic Equation Calculator examples?
These examples turn the Quadratic Equation Calculator inputs into complete scenarios with stated values, calculation steps, and interpreted results. Use them to check how the calculator behaves, compare your own situation with realistic cases, and identify which inputs have the greatest effect before relying on an estimate.
Example 1: Two Real Roots
x² - 3x + 2 = 0
- Identify a = 1, b = -3, and c = 2.
- Calculate D = (-3)² - 4(1)(2) = 1.
- Substitute x = (3 ± 1)/2.
- The roots are x = 1 and x = 2.
The vertex is (1.50, -0.25), and the upward-opening parabola crosses the x-axis at both roots.
Example 2: One Repeated Root
x² + 2x + 1 = 0
- Identify a = 1, b = 2, and c = 1.
- Calculate D = 2² - 4(1)(1) = 0.
- Both formula branches give x = -1.
The vertex is (-1, 0), so the parabola touches the x-axis once without crossing it.
Example 3: Complex Conjugate Roots
x² + 2x + 5 = 0
- Identify a = 1, b = 2, and c = 5.
- Calculate D = 2² - 4(1)(5) = -16.
- Use √(-16) = 4i.
- The roots are -1 + 2i and -1 - 2i.
The vertex is (-1, 4), so the upward-opening parabola has no real x-intercepts.
Example 4: Non-Monic Equation
2x² + 5x - 3 = 0
- Calculate D = 5² - 4(2)(-3) = 49.
- Substitute x = (-5 ± 7)/4.
- The roots are -3 and 0.50.
Their sum is -2.50 = -b/a and their product is -1.50 = c/a, confirming both solutions.
What the Examples Show
For “What the Examples Show,” the Quadratic Equation Calculator examples organizes the relevant inputs, applies the documented method, and shows a result that can be compared across scenarios. The points below identify the specific considerations and assumptions users should review before relying on the estimate.
- A positive discriminant produces two graph intersections.
- A zero discriminant places the repeated root at the vertex.
- A negative discriminant produces complex roots and no real x-intercepts.
- The quadratic formula works whether a equals 1 or another nonzero value.
- Vieta's formulas provide a quick independent root check.
Enter each scenario in the Quadratic Equation Calculator and compare the algebraic results with the graph.
Applied-Unit Reminder
The solver accepts algebraic coefficients rather than metric or imperial inputs. If an equation models a physical path, area, cost, or engineering quantity, convert all measurements to one consistent unit before deriving a, b, and c.
Corrections and Review
Report an issue or error in the calculator using the Contact Us page.