Convert a quadratic expression
Enter coefficients for a quadratic in the form ax² + bx + c. The result appears above this form after calculation.
Formula used
The calculator first evaluates the discriminant. It then identifies roots and writes the matching factors.
r₁, r₂ = (−b ± √D) / (2a)
ax² + bx + c = a(x − r₁)(x − r₂)
When integer binomials exist, the calculator searches for factors that multiply back to the original standard form.
How to use this calculator
- Enter the values of a, b, and c from the standard quadratic form.
- Choose the variable letter and the precision needed for displayed roots.
- Select Convert to factored form to show the result above the input form.
- Check the discriminant and root type before interpreting intercepts or solutions.
- Expand the result manually when you need a final algebra check.
Example data table
| Standard form | Factored form | Roots |
|---|---|---|
| x² − 5x + 6 | (x − 2)(x − 3) | 2, 3 |
| 2x² + 7x + 3 | (2x + 1)(x + 3) | −1/2, −3 |
| x² + 4x + 4 | (x + 2)² | −2, −2 |
| x² + 2x + 2 | [x − (−1 + i)] [x − (−1 − i)] | −1 + i, −1 − i |
Why factored form matters
Converting standard form to factored form helps reveal a quadratic’s zeros. Standard form usually appears as ax² + bx + c. Factored form shows the expression as products of linear factors. Each factor can identify a value that makes the quadratic equal zero. This calculator performs the conversion and explains the related values.
A quadratic can factor cleanly when its roots are rational. For example, x² − 5x + 6 becomes (x − 2)(x − 3). Multiplying the factors returns the original expression. The first terms create x². The outer and inner terms combine to −5x. The constants create positive six. This quick check confirms the result.
Not every quadratic produces whole-number factors. Some expressions have fractional roots. Others include square roots. The discriminant decides which situation applies. It comes from b² − 4ac. A positive perfect square often gives rational roots. A positive non-square gives irrational roots. Zero gives one repeated root. A negative discriminant produces complex roots.
The calculator accepts decimal or integer coefficients. Enter the number multiplying the squared variable first. Then enter the linear coefficient. Finally, enter the constant. You can select a variable letter and choose display precision. After calculation, the result area shows the standard expression, discriminant, roots, factorization, and a verification statement.
Integer factorization is the clearest result when it exists. The calculator looks for a common factor first. It then checks possible binomial pairs. This approach can produce answers such as 2(x + 1)(x + 3). Keeping a common factor outside makes the expression easier to read. It also prevents equivalent but less convenient results.
When clean integer binomials do not exist, root form still provides a correct factorization. The expression can be written as a(x − r₁)(x − r₂). Here, r₁ and r₂ are roots from the quadratic formula. This form works for rational, irrational, and complex answers. It connects factoring directly with solving equations.
Use signs carefully when reviewing factors. A factor written as x − 4 has a root of four. A factor written as x + 4 has a root of negative four. The leading coefficient also matters. It remains outside the two root factors unless it is absorbed into binomials. Expanding your final result is the best safeguard against sign errors.
Factored form supports graphing and equation solving. The roots locate x-intercepts when the graph is real. Repeated factors show a graph that touches the axis. Two different real factors show separate intercepts. Complex factors have no real x-intercepts. These features make factorization useful beyond classroom exercises.
For best results, enter coefficients exactly whenever possible. Fractions can be entered as decimals, although repeating values may round. Use more display precision for close roots. Review the discriminant before interpreting the factors. A calculator result is a guide, but expansion remains the final proof. Careful checking builds durable algebra skills.
Careful use also reveals coefficient patterns. They often help with later algebra problems.
Frequently asked questions
1. What is standard form for a quadratic?
Standard form is ax² + bx + c, where a is not zero. It groups terms by descending powers and makes the coefficients easy to identify.
2. What is factored form?
Factored form writes the quadratic as products, often a(x − r₁)(x − r₂). The roots are the values that make the expression equal zero.
3. Can this calculator use decimal coefficients?
Yes. Enter decimals in any coefficient field. The calculator evaluates the discriminant and displays roots using your selected decimal precision.
4. What does the discriminant show?
The discriminant, b² − 4ac, identifies the root pattern. Positive values give two real roots. Zero gives a repeated root. Negative values give complex roots.
5. Why are my factors not whole numbers?
Some quadratics do not have integer roots. Their correct factors may include fractions, radicals, or complex values. The factorization remains valid.
6. What happens when a equals zero?
The expression becomes linear rather than quadratic. The calculator still gives a linear factor when possible, but quadratic discriminant rules no longer apply.
7. Why does x + 4 give a root of negative four?
Set the factor equal to zero: x + 4 = 0. Subtracting four gives x = −4. Factor signs are opposite their matching roots.
8. How can I verify a factored answer?
Multiply the factors. Combine like terms. A correct expansion recreates the original ax² + bx + c expression exactly.
9. Does a repeated factor matter?
Yes. A repeated factor such as (x − 3)² means the quadratic has one root repeated twice. On a real graph, it touches the x-axis there.
10. Can I change the variable letter?
Yes. Enter one alphabetic character in the variable field. The calculator uses that letter in the standard form, factors, formulas, and examples.
11. Why should I use more decimal precision?
Extra precision helps when roots are close together or irrational. It reduces display rounding, though full precision should be used for final verification.