تبسيط العبارات الجبرية والتحليل وحل المعادلات

Let's solve this mathematics exercise from the 2008 BEM exam. We have two parts. In the first part, we need to expand and simplify the expression A.

Part one asks us to expand A, which is the square of two minus the square root of three.

We use the identity: a minus b squared equals a squared plus b squared minus two times a times b.

Applying this, we get two squared, minus two times two times the square root of three, plus the square root of three squared.

Simplifying, two squared is four, and the square root of three squared is three. Four plus three is seven. So, A equals seven minus four times the square root of three.

Thus, the simplified form of A is seven minus four root three.

Now for part two. We are given an algebraic expression E, which is x squared minus the quantity seven minus four root three.

In the first sub-question, we need to calculate the exact value of E when x equals root seven.

Substitute square root of seven for x. We get root seven squared minus the parenthetical expression.

The square root of seven squared is simply seven. Notice that seven minus seven will cancel out.

Distribute the negative sign. Seven minus seven is zero, and negative times negative becomes positive. So E equals four square root of three.