Quadratic expressions

Here (4x^2=(2x)^2) and (9=3^2), so it is a difference of two squares. Exam tip: apply the (a^2-b^2) identity quickly.

The direct answer is option B: \((x-1)^2\). Use the identity \((a-b)^2=a^2-2ab+b^2\). Taking \(a=x\) and \(b=1\), we get \((x-1)^2=x^2-2x+1\), exactly the given expression. Option A, \((x+1)^2\), expands to \(x^2+2x+1\), so its middle term has the wrong sign. Option B is correct because its expansion gives all three terms, including \(-2x\). Option C, \((x-2)^2\), expands to \(x^2-4x+4\); both the middle coefficient and constant are wrong. Option D, \(x^2-1\), is a difference of squares, but it is not equal to the given expression; it has no \(-2x\) term and has constant \(-1\). The reliable method is to compare the middle term and the last term: the last term 1 suggests the number 1, and the negative middle term tells us to use subtraction. Memory cue: plus inside a square gives a plus middle term, while minus inside gives a minus middle term.

The main idea is substitution: when a value of a variable is given, replace the variable with that value everywhere in the expression. Here, every x becomes 2. The expression therefore becomes a numerical calculation, not an expression with an unknown value. The order of operations is important because the square is evaluated before multiplication and addition or subtraction.

Substitute and calculate: \(x^2+5x-6=2^2+5(2)-6\). First, \(2^2=4\); next, \(5(2)=10\). Thus the result is \(4+10-6=8\). Therefore option A, 8, is correct. Option B would result from forgetting the subtraction, while the other choices do not follow from the substitution.