Question

Multiply. $$(2 x-3)^{2}$$

Short answer

\(4x^{2} - 12x + 9\)

Step-by-step solution

Expand the Expression

To begin with, expand the expression \((2x - 3)^2\) using the formula \((a-b)^2 = a^2 - 2ab + b^2)\n\n\Substituting a = 2x and b = 3, we have \((2x)^{2} - 2(2x)(3) + (3)^{2}\.

Simplify the Expression

Now, simplify the expression.\n\n \Solve \((2x)^{2}\) to get \(4x^{2}\), then multiply \(2\) by \(2x\) and \(3\) to get \(12x\), and square \(3\) to get \(9\).\n\n \Now the expression: \(4x^{2} - 12x + 9'\)

Final Result

The final result is no more simplifiable. \n\nTherefore, \((2x - 3)^{2}\) simplifies to \(4x^{2} - 12x + 9\)

Key concepts

Binomial Squares

Simplifying a binomial square is a common exercise in algebra that students come across. The term 'binomial squares' refers to squaring a binomial expression, which consists of two terms. The general formula for a binomial square is \(a+b)^2 = a^2 + 2ab + b^2\). Similarly, \(a-b)^2 = a^2 - 2ab + b^2\) for the square of a difference between two terms.

When you come across an expression like \(2x-3)^2\), it's a clear signal to apply this formula. Remembering these formulas can greatly simplify the process of expanding binomial squares. A common mistake to avoid when dealing with binomial squares is to incorrectly square the individual terms without applying the middle product (the \(-2ab\) or \(+2ab\) part), which is critical for the correct expansion.

Polynomial Expansion

Polynomial expansion involves the process of spreading out a polynomial expression that has been raised to a power, such as a binomial square or higher powers. It requires the proper application of algebraic laws, such as the distributive property, where every term in the first polynomial must multiply every term in the second polynomial.

Expanding \(2x - 3)^2\) is a clear example of polynomial expansion. It's not just squaring each term individually; rather, it's more like using the FOIL (First, Outer, Inner, Last) technique for the first and second binomials in a multiplication. For learners, visualizing the expansion with the FOIL method, or writing out each step, can often make polynomial expansion clearer and less prone to simple mistakes.

Algebraic Manipulation

Algebraic manipulation is the technique of rearranging and simplifying algebraic expressions to either solve for a variable or to present the expression in a more understandable form. It encompasses operations such as expanding, factoring, canceling, and combining like terms.

For example, after expanding \(2x - 3)^2\), you end up with \(4x^2 - 12x + 9\). Algebraic manipulation can then be used to further simplify expressions if possible, by combining like terms or factoring. In this case, no further simplification is possible, but in other cases, you might be able to simplify the expression significantly. This step is where you finalize your answer, ensuring that it's in its simplest form before presenting it.