Calculator
Graph
Example Data Table
| a | b | c | Standard Form | Factored Form |
|---|---|---|---|---|
| 1 | -5 | 6 | x² - 5x + 6 | (x - 2)(x - 3) |
| 1 | 2 | 1 | x² + 2x + 1 | (x + 1)² |
| 2 | -8 | 6 | 2x² - 8x + 6 | 2(x - 1)(x - 3) |
Formula Used
Standard quadratic form is:
ax² + bx + c
The discriminant is:
D = b² - 4ac
The roots are:
x = (-b ± √D) / 2a
The factored form is:
a(x - r1)(x - r2)
How to Use This Calculator
Enter the values of a, b, and c from your standard form equation.
Press the convert button.
The calculator finds the discriminant, roots, vertex, y-intercept, and factored form.
You can also download the answer as CSV or PDF.
Standard to Factored Form Conversion Guide
What This Calculator Does
This calculator changes a quadratic equation from standard form into factored form. Standard form is written as ax² + bx + c. Factored form shows the roots directly. It is useful when solving equations, graphing parabolas, and checking x-intercepts.
Why Factored Form Matters
Factored form helps you see where the graph crosses the x-axis. These crossing points are called roots or zeros. When the roots are real, the expression can be written as a product of linear factors. This makes many algebra problems easier.
Discriminant Check
The calculator first finds the discriminant. A positive discriminant gives two real roots. A zero discriminant gives one repeated root. A negative discriminant gives complex roots. In that case, the graph does not cross the x-axis.
Advanced Result Details
The tool also shows the vertex and y-intercept. The vertex is the turning point of the parabola. The y-intercept is found from c. These values help you understand the full shape of the quadratic graph.
Graph Support
The graph gives a visual check of the equation. You can compare the curve with the roots and intercepts. This helps students understand how algebra and geometry connect. It also helps detect input errors quickly.
Practical Use
Use this calculator for homework, teaching, test review, and quick algebra checks. It works best when you know the coefficients from standard form. Enter accurate values. Then review the roots and factored form carefully.
FAQs
What is standard form?
Standard form is ax² + bx + c, where a, b, and c are numbers. The value of a should not be zero for a quadratic equation.
What is factored form?
Factored form writes a quadratic as a product, usually a(x - r1)(x - r2). It shows the roots clearly.
What are roots?
Roots are x-values that make the equation equal zero. They are also called zeros or x-intercepts when real.
What does the discriminant show?
The discriminant shows the root type. Positive means two real roots. Zero means one repeated root. Negative means complex roots.
Can every quadratic be factored?
Every quadratic can be factored over complex numbers. Only some can be factored neatly using real numbers or integers.
Why is the coefficient a included?
The value of a controls the width and direction of the parabola. It also stays outside the root factors.
What happens when a is zero?
When a is zero, the expression is not quadratic. The calculator treats it as a linear expression if possible.
Does this calculator show steps?
Yes. It shows the discriminant method, root formula, and final factored structure for clear understanding.