Quadratic expressions

A binomial square has the form \(a^2-2ab+b^2\). With \(a=2x\) and \(b=3y\), the middle term is \(-2ab=-12xy\), so A is \((2x-3y)^2\). In B, the last term should be \(9y^2\), not \(6y^2\). Exam tip: verify all three terms.

Here, \(49x^2=(7x)^2\) and \(81y^2=(9y)^2\). Using \(a^2-b^2=(a+b)(a-b)\) gives option A. Squared options produce a middle term. Exam tip: first check whether both terms are perfect squares.

With identical linear factors, the expression has the form \((x+r)^2=x^2+2rx+r^2\). Thus \(p=2r\) and \(q=r^2\), giving \(p^2=4q\). Exam tip: compare the middle coefficient with \(2r\) when spotting a perfect square.

\(9a^2-12ab+4b^2=(3a-2b)^2=(3a-2b)(3a-2b)\), so it is the product of two identical linear factors. Here \(9a^2=(3a)^2\), \(4b^2=(2b)^2\), and the middle term is \(2\times3a\times(-2b)=-12ab\). In option B, the last term is negative, whereas the square of a linear binomial has a positive last square term. Exam tip: for a perfect-square trinomial, take the square roots of the first and last terms and verify the middle term using \(\pm2pq\).

For (x^2+8x+18), the constant should be (16) for a perfect square. In exams, check the square of half the middle coefficient.

\(x^2+6x+9=x^2+2\cdot x\cdot3+3^2=(x+3)^2\), so it is the product of two identical binomials. \(x^2+6x+8=(x+2)(x+4)\), whose factors differ. Exam tip: in a perfect square, the middle term is \(2ab\).

((x+3)^2=x^2+6x+9), so the given expression is not (1) less but (4) less. The correct comparison should be (4) less.

To make (x^2+6x) a perfect square, (9) is needed, and (5) is already present, so add (4). In exams, square half the middle coefficient.

For (x^2-8x), the perfect square constant should be (16), so add (16-10=6). In exams, subtract the current constant.

A quadratic expression must have highest power of \(x\) equal to 2, so the coefficient \(a\) of \(x^2\) must be non-zero. Both \(b\) and \(c\) may be zero. Exam tip: check the coefficient of the highest power first.

The first is a difference of squares and the second is a perfect square. In exams, identify by the presence of the middle term.

For an expression to be quadratic, the coefficient of \(x^2\) must not be zero. In A, \(a^2+1>0\) for every real a, so it is always quadratic. In B, taking \(a=0\) makes it linear. Exam tip: check the \(x^2\) coefficient first.

(16=4^2), and the negative middle term is (-2\cdot x\cdot4=-8x). In exams, choose the coefficient according to the given sign.

A perfect-square trinomial has zero discriminant, so \(b^2=4ac\). Substituting \(c=\frac{b^2}{4a}\) gives \(a\left(x+\frac{b}{2a}\right)^2\). For this to be a square of a real linear expression, \(a\) must be positive. Exam tip: check the sign of \(a\) along with the discriminant.

For \((px+q)^2\), we have \(a=p^2,\ b=2pq\), and \(c=q^2\). Thus \(b^2-4ac=(2pq)^2-4p^2q^2=0\). A positive discriminant gives distinct real roots, not a repeated root. Exam tip: test a perfect-square quadratic using its discriminant.

Using \(a^2-b^2=(a-b)(a+b)\), put \(a=p\) and \(b=q\) to get \(p^2-q^2=(p-q)(p+q)\). In \((p-q)^2\), the middle term is \(-2pq\), so it is different. Exam tip: look for a subtraction between two squares.

The expression must be \((x\pm4)^2\). Expanding gives \((x+4)^2=x^2+8x+16\) and \((x-4)^2=x^2-8x+16\), so \(p=\pm8\). Exam tip: take the square root of the constant term and check the middle term.

In a quadratic expression, the coefficient of \(x^2\) must be non-zero; hence \(a\ne0\). The values of \(b\) or \(c\) may be zero. Exam tip: check the non-zero coefficient of the highest power.

In \((x-3)(2x+1)\), the highest-degree product is \(x\times2x=2x^2\), so its degree is 2 and it is quadratic. B and C have degree 3, while D contains \(x^{-1}\), so it is not a polynomial. Exam tip: check the highest power first.

\(x^2+4x+5=(x+2)^2+1\). Since \((x+2)^2\geq0\), its minimum value is 1, so it is positive for every real \(x\). In the closest distractor, option B, substituting \(x=2\) gives \(-1\), so it is not always positive. Exam tip: complete the square to find the minimum or maximum value of a quadratic.