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Which of the following expressions is a perfect-square trinomial and can be factorised as the square of a binomial?
Correct answer: A
A perfect-square trinomial has the form \(a^2-2ab+b^2=(a-b)^2\). Here, \(4x^2=(2x)^2\), \(9y^2=(3y)^2\), and the middle term is \(-2(2x)(3y)=-12xy\). Therefore, \(4x^2-12xy+9y^2=(2x-3y)^2\). In option B, the last term would need to be \(9y^2\); in option C, the middle term would need to be \(-12xy\). Exam tip: take the square roots of the first and last terms, then verify the middle term using \(\pm2ab\).
Which of the following expressions can be factorised into two identical binomial factors?
Correct answer: A
In \(x^2+10x+25\), \(25=5^2\) and the middle term is \(2\times x\times5=10x\). Hence it equals \((x+5)^2\). In \(x^2+10x+24\), the constant term is not 25. Exam tip: verify the middle term for a perfect square.
The expression is a quadratic trinomial. To factor it, we need two binomials whose first terms multiply to give the leading term, whose last terms multiply to give the constant term, and whose cross-products add to the middle term. This is a useful way to check every proposed factorisation rather than relying only on appearance.
For choice A, expand \\(2x+3\\)(5x+8). The products are \\(10x^2\\), \\(16x\\), \\(15x\\), and \\(24\\). The middle terms add to \\(31x\\), so the result is exactly \\(10x^2+31x+24\\). Therefore A is correct. The other choices do not produce the required middle coefficient or constant term.
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