Expand two binomials of the form (a₁x + a₀)(b₁x + b₀) and see step-by-step solution.
This calculator expands two binomials of the form (a₁x + a₀)(b₁x + b₀) using the distributive (FOIL) method. It provides the expanded quadratic expression instantly and optionally shows a step-by-step solution.
Automatically multiplies each term in the first binomial by each term in the second binomial.
Resulting in a quadratic expression of the form:
C₂x² + C₁x + C₀
Enable “Show Step-by-Step” to see each part of the expansion:
If the coefficient a₁ = 0, the expression is no longer quadratic and the result is linear. The calculator warns you automatically.
Inputs have clear placeholders and ARIA labels, making it beginner-friendly. Adjust the number of decimals using the precision dropdown for cleaner results.
1️⃣ Enter the coefficients and constants of the two binomials:
2️⃣ The calculator performs the FOIL multiplication:
3️⃣ If "Show Step-by-Step" is enabled, each multiplication and combination step is displayed for clarity.
4️⃣ If a₁ = 0, the calculator alerts you that the result is a linear expression.
Multiplying two binomials involves combining each term from the first binomial with every term from the second binomial. Mathematically, if we have two binomials of the form (a₁x + a₀) and (b₁x + b₀), the product is a quadratic expression:
(a₁x + a₀)(b₁x + b₀) = a₁b₁x² + (a₁b₀ + a₀b₁)x + a₀b₀
In this formula:
This method is commonly referred to as the FOIL method, which stands for First, Outer, Inner, Last. It is an efficient way to ensure that every term is multiplied properly and no terms are missed. FOIL is especially useful for beginners learning algebra because it breaks the multiplication into clear, manageable steps.
Step-by-Step Example: Consider multiplying (3x - 2)(x + 5):
Final Result: 3x² + 13x - 10
This formula works for both positive and negative coefficients, as well as decimal numbers. By understanding the binomial multiplication formula, you can easily expand, simplify, or factor expressions in algebra. It is also foundational for higher-level mathematics, such as polynomial multiplication, quadratic equations, and calculus operations.
Some key points to remember:
Overall, the binomial multiplication formula is a powerful tool for algebraic calculations, allowing students and professionals alike to quickly expand expressions and verify their results. It serves as a stepping stone to understanding polynomials, factoring, quadratic equations, and more advanced mathematical operations.
If you enter:
a₁ = 3, a₀ = -2, b₁ = 1, b₀ = 5
The calculator expands:
3x² + 13x - 10
Step-by-step (if enabled):
1️ 3 * 1 = 3x²
2️⃣ (3 * 5) + (-2 * 1) = 13x
3️⃣ (-2 * 5) = -10
4️⃣ Combine: 3x² + 13x - 10
Everything you need to know about multiplying two binomials, expanding expressions, and understanding step-by-step solutions.
It multiplies two binomials of the form (a₁x + a₀)(b₁x + b₀) using the distributive (FOIL) method and provides the expanded quadratic expression instantly. You can also view each step of the expansion if needed.
FOIL stands for First, Outer, Inner, Last. It is a technique to multiply two binomials:
• First: Multiply the first terms of each binomial
• Outer: Multiply the outer terms
• Inner: Multiply the inner terms
• Last: Multiply the last terms
Then combine like terms to get the final expanded expression.
If a₁ = 0, the expression is no longer quadratic. The calculator will automatically display a warning, and the resulting expression is linear, of the form C₁x + C₀.
Yes. Enable the “Show Step-by-Step” checkbox to see each part of the calculation: multiplying first terms, outer & inner terms, last terms, and then combining like terms to form the final expanded expression.
The precision dropdown lets you choose how many decimal places the results display. This is useful if your coefficients are decimals and you want a cleaner, more readable output.
Yes. The calculator accepts positive, negative, and decimal numbers for all coefficients and constants, and will compute the expanded quadratic expression accurately.
Yes! It is completely free to use, requires no registration, and works instantly on any device with an internet connection.
Absolutely. The calculator is fully responsive and works smoothly on smartphones, tablets, laptops, and desktop computers.
The result is a quadratic expression of the form C₂x² + C₁x + C₀. You can use it to factor further, graph, or solve the quadratic equation if needed.
Yes. After performing all multiplications, the calculator automatically combines like terms to provide the final simplified quadratic expression.
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