Question Video: Expanding and Simplifying the Product of Two Binomials in a Real-Life Context Mathematics

Video Transcript

If a triangular plot of land has a base of two 𝑥 plus two meters and a height of negative 𝑥 plus five meters, which of the following expressions represents the area of the plot of land? (A) Negative two 𝑥 squared plus eight 𝑥 plus 10 square meters. (B) Negative two 𝑥 squared plus 12𝑥 plus 10 square meters. (C) Two 𝑥 squared plus 12𝑥 plus 10 square meters. (D) 𝑥 squared plus six 𝑥 plus five square meters. (E) Negative 𝑥 squared plus four 𝑥 plus five square meters.

We’ve been given algebraic expressions for the base and height of this triangular plot of land and asked to determine which of five given expressions represents its area. We know that the area of a triangle is calculated using the formula base multiplied by height over two. So the area of this triangle is equal to two 𝑥 plus two multiplied by negative 𝑥 plus five all over two, and we’ll include the units at the end. We now need to simplify this expression.

First, we note that we can take a factor of two out of the first linear factor so that the expression becomes two multiplied by 𝑥 plus one multiplied by negative 𝑥 plus five all over two. We can then cancel a factor of two from the numerator and denominator to leave 𝑥 plus one multiplied by negative 𝑥 plus five.

The next step in simplifying this expression is to distribute the parentheses. When we multiply two binomials together, we should initially obtain four terms, found by multiplying each term in the first binomial by each term in the second. We need to do this in a systematic way to ensure we multiply each pair of terms together.

Multiplying the first terms in each binomial together gives negative 𝑥 squared. Then, multiplying the terms on the outside together, so that’s the first term in the first binomial and the second term in the second binomial, gives positive five 𝑥. Multiplying the terms on the inside of the product together gives negative 𝑥. And finally, multiplying the last terms in each binomial together gives positive five. And we have the four terms we were expecting.

We can simplify this expression one stage further by combining the like terms of five 𝑥 and negative 𝑥 to give positive four 𝑥. We’ve therefore found that an expression for the area of the triangular plot of land is negative 𝑥 squared plus four 𝑥 plus five square meters, which is option (E).