Quadratic expressions

In a quadratic expression, the coefficient of the \(x^2\) term must not be zero. If \(a=0\), the term \(ax^2\) disappears and the expression becomes \(bx+c\), which may be linear (or constant when \(b=0\)), not quadratic. Option C refers to a linear expression with highest power 1. Exam tip: The degree of a polynomial is determined by the highest power with a non-zero coefficient.

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

A polynomial whose greatest exponent of x is 2 has degree 2, so it is called a quadratic polynomial. A linear polynomial has highest power 1. Exam tip: identify a polynomial’s type by its greatest exponent.

Using \((a+b)^2=a^2+2ab+b^2\), put \(a=x\) and \(b=5\) to get \(x^2+10x+25\). Thus, the \(10x\) middle term is missing. Exam tip: always check the \(2ab\) term.

Using the distributive property, \((2x+3)(x-2)=2x\cdot x+2x\cdot(-2)+3\cdot x+3\cdot(-2)\). Thus, \(2x^2-4x+3x-6=2x^2-x-6\). Therefore, option D is correct. In option C, the constant term is incorrectly written as \(+6\) instead of \(-6\). Exam tip: multiply each term of one binomial by both terms of the other, then combine like terms.

Using the distributive property, \((3x-1)(x+4)=3x\cdot x+3x\cdot4-1\cdot x-1\cdot4=3x^2+12x-x-4=3x^2+11x-4\). Therefore, option D is correct. In option A, \(12x-x\) has incorrectly been simplified to \(x\). Exam tip: after expanding, check the sign of every product before combining like terms.

\(9p^2-25q^2=(3p)^2-(5q)^2\), so it is a difference of two squares. Its factorisation is \((3p-5q)(3p+5q)\). Option C is \((3p-5q)^2\), which is a perfect square, not a difference of squares. Exam tip: when two squared terms have a minus sign and no middle term, check the form \(a^2-b^2\).

\(a^2+10a+25=(a+5)^2\) because the middle term is \(2\times a\times5=10a\). In option B, the constant 20 is not \(5^2=25\). Exam tip: use the square roots of the first and last terms to check the middle term.

Half the coefficient in (-10x) is (-5), and its square is (25). Exam tip: square half of the coefficient.

A quadratic expression has highest power (2). Exam tip: identify the type by checking the power.

A quadratic expression must have 2 as the highest power of \(x\), so the coefficient \(a\) of \(x^2\) cannot be zero. For example, \(3x^2-5x+1\) is quadratic. Both \(b\) and \(c\) may be zero. Exam tip: check the non-zero coefficient of the highest power first.

A quadratic expression has highest power 2, so the coefficient of \(x^2\) must be non-zero. The \(x\)-term and constant may be zero. Exam tip: identify the highest non-zero power first.

A quadratic expression must have 2 as the highest power of \(x\), so the coefficient \(a\) of \(x^2\) cannot be zero. If \(a=0\), the \(x^2\) term disappears and the expression becomes linear or constant. Exam tip: check the highest non-zero power.

The conjugate binomials \((p+q)\) and \((p-q)\) give \((p+q)(p-q)=p^2-q^2\), as the middle terms cancel. Option B is a binomial square. Exam tip: opposite signs with matching terms indicate a difference of squares.

Using \((a-b)^2=a^2-2ab+b^2\), with \(a=x\) and \(b=5\), gives \((x-5)^2=x^2-2(x)(5)+25=x^2-10x+25\). Hence, option B is correct. Option C has an incorrect positive middle term, while option D is the expansion of \((x-5)(x+5)\), not \((x-5)^2\). Exam tip: in \((a-b)^2\), the middle term is always negative, \(-2ab\).

Here (36=6^2) and (12x=2\cdot6\cdot x). Exam tip: confirm using the square root of the last term.

Here (49=7^2) and (-14x=-2\cdot7\cdot x). Exam tip: identify ((x-7)^2) from the negative middle term.

A quadratic expression must have a non-zero coefficient of \(x^2\), so \(a\ne0\) is necessary. If \(a=0\), the highest possible power becomes 1. Exam tip: first check the coefficient of the squared term.

A quadratic expression has highest variable power 2, with non-negative integer powers. In \(7y^2-3y+4\), the highest power is 2. Option B is cubic. Exam tip: check the greatest exponent first.

The linear term is (0x), so its coefficient is (0). Exam tip: take the missing (x)-term coefficient as (0).