Squaring Binomials in Equations
Squaring a binomial in an equation means applying the identity (a + b) squared = a squared + 2ab + b squared, or (a - b) squared = a squared - 2ab + b squared, before solving. This arises in Chapter 9 of OpenStax Elementary Algebra 2E when solving radical equations — you square both sides to eliminate a radical, producing a quadratic that may require factoring. The middle term 2ab is what students most often forget, turning the square of a binomial into just a squared plus b squared. Correctly expanding the binomial square is essential for accurate solutions.
Key Concepts
Property When squaring a binomial, use the binomial squares formulas. Don't forget the middle term! $$(a+b)^2 = a^2 + 2ab + b^2$$ $$(a b)^2 = a^2 2ab + b^2$$.
Examples Solve $\sqrt{x 2} + 2 = x$. First, isolate the radical: $\sqrt{x 2} = x 2$. Squaring both sides gives $x 2 = (x 2)^2 = x^2 4x + 4$. The solutions are $x=2$ and $x=3$.
Solve $\sqrt{k+7} = k 5$. Squaring both sides gives $k+7 = (k 5)^2 = k^2 10k + 25$. This simplifies to $0 = k^2 11k + 18$, which factors to $(k 2)(k 9)$. Only $k=9$ is a valid solution.
Common Questions
How do you square a binomial?
(a + b) squared equals a squared + 2ab + b squared. The middle term 2ab is critical and is often forgotten. For example, (x + 3) squared = x squared + 6x + 9.
What is the binomial square identity?
There are two: (a + b) squared = a squared + 2ab + b squared, and (a - b) squared = a squared - 2ab + b squared. Both follow the same pattern; only the sign of the middle term differs.
Why does squaring arise when solving radical equations?
To eliminate a square root, you square both sides of the equation. If one side contains a binomial after the radical is isolated, you must square the entire binomial correctly using the identity.
What is the most common mistake when squaring a binomial?
Writing (a + b) squared as a squared + b squared, omitting the 2ab middle term. This is incorrect. Always expand fully using FOIL or the binomial square formula.
When do students learn squaring binomials in equations?
This concept appears in algebra 1 when solving radical equations, covered in OpenStax Elementary Algebra 2E Chapter 9: Roots and Radicals.
Can squaring both sides introduce extraneous solutions?
Yes. Squaring is not a reversible operation, so it can create solutions that do not satisfy the original equation. Always check solutions in the original equation after squaring.
How do I expand (2x - 5) squared?
(2x - 5) squared = (2x) squared - 2(2x)(5) + 5 squared = 4x squared - 20x + 25.