Can you factor when Δ<0?

In real numbers, Δ<0 means no real roots, so the polynomial cannot be factored with real coefficients. Complex factoring is possible.

What is cross-multiplication factoring?

Split ac into two numbers whose sum equals b. For x²+5x+6: 6=2×3 and 2+3=5, so (x+2)(x+3).

What is factoring used for?

Factoring simplifies expressions, solves equations, finds function zeros, reduces fractions, and proves algebraic identities.

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Home›Algebra Tools›Factorization Calculator

Factorization Calculator

Enter coefficients a, b, c of ax²+bx+c to factor into two linear factors

a =

b =

c =

Factorization Formula

ax²+bx+c = a(x-x₁)(x-x₂), x₁₂=(-b±√Δ)/2a, Δ=b²-4ac

Find roots x₁, x₂ using the quadratic formula, then apply the factorization. If Δ<0, real factoring is not possible.

⚠Note: a cannot be 0 (not quadratic). If Δ<0, cannot factor in real numbers. If Δ=0, it's a perfect square.

What is Factorization?

Factorization rewrites a polynomial as a product of simpler polynomials (factors). It is one of the most important algebraic transformation techniques.

Quadratic Formula

Find roots x₁=(-b+√Δ)/2a, x₂=(-b-√Δ)/2a, then substitute into a(x-x₁)(x-x₂).

Cross-Multiply

Split ac into two factors whose sum is b. x²+5x+6: 6=2×3, 2+3=5, so (x+2)(x+3).

Discriminant

Δ>0 two distinct factors; Δ=0 perfect square; Δ<0 no real factoring.

Perfect Square

When Δ=0, e.g. x²-4x+4 = (x-2)², both factors are identical.

💡 Example: Factor x²+5x+6. Δ=25-24=1>0, x₁=(-5+1)/2=-2, x₂=(-5-1)/2=-3, so x²+5x+6 = (x+2)(x+3).

Applications

Equation Solving Function Zeros Simplification Proofs Fraction Ops Competitions

FAQs about Factorization

What is a linear factor?▼

A polynomial of degree 1 like (x+1), (2x-3), (3x+5). A quadratic factors into two linear factors.

Can factor when Δ<0?▼

Not in real numbers. With complex numbers, it factors as (x-x₁)(x-x₂) where x₁,x₂ are complex conjugates.

Cross-multiply method?▼

Split constant c into m×n where m+n=b. x²+5x+6: 6=2×3, 2+3=5 → (x+2)(x+3).

Why factor?▼

Essential for solving higher-degree equations, simplifying expressions, reducing fractions, and proving identities.

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