Quadratic Expressions

What terms, coefficients and constant terms are; how to tell a linear expression from a quadratic one; and how to add, subtract and simplify expressions with $x^2$ terms.

Before you start

In Secondary 1 you learnt that an algebraic expression is made of letters and numbers joined by $+$ , $-$ , $\times$ and $\div$ , such as $3x + 5$ or $\frac{6 - x}{2}$ , and that like terms have the same letters, so they can be added: $3x + 5 - x + 2 = 2x + 7\text{.}$

This chapter adds terms such as $x^2$ and $xy$ . This lesson gives them their names and shows how to collect them.

Terms, Coefficients and Constant Terms

The parts of an expression joined by $+$ and $-$ are its terms. Each term takes the sign in front of it. In $x - 3xy + 4y - 7$ the terms are $x$ , $-3xy$ , $4y$ and $-7$ .

Coefficients and the constant term

The coefficient of a term is the number that multiplies its letters, together with its sign. In $x - 3xy + 4y - 7$ the coefficient of $xy$ is $-3$ and the coefficient of $y$ is $4$ .

A letter on its own has coefficient $1$ , and $-x$ has coefficient $-1$ : $x = 1 \times x$ and $-x = -1 \times x$ .

A term with no letter is the constant term. The constant term of $x - 3xy + 4y - 7$ is $-7$ .

Example 1: Naming the parts

For the expression $5 - 3x^2 + x$ , write down (a) its terms, (b) the coefficient of $x^2$ , (c) the coefficient of $x$ , (d) the constant term. Then (e) write down the coefficient of $x$ in $x^2 - x$ .

Solution

(a) Each term takes the sign in front of it: the terms are $5$ , $-3x^2$ and $x$ .

(b) The number multiplying $x^2$ is $-3$ , sign included.

(d) The term with no letter is $5$ .

(e) In $x^2 - x$ , the term in $x$ is $-x = -1 \times x$ , so the coefficient is $-1$ .

Terms $5$ , $-3x^2$ , $x$ ; coefficient of $x^2$ is $-3$ ; coefficient of $x$ is $1$ ; constant term $5$ . In $x^2 - x$ the coefficient of $x$ is $-1$ .

What $x^2$ , $-4x^2$ and $xy$ Mean

Just as $3 \times 3 = 3^2$ , we write $x \times x$ as $x^2$ , read " $x$ squared". It is not $2x$ , which is $x + x$ . When $x = 3$ , $x^2 = 9$ but $2x = 6$ .

Notation

$x^2 = x \times x$ , and $2x = 2 \times x = x + x$ .

$-4x^2 = -4 \times x \times x$ : the square belongs to $x$ only, not to the $-4$ .

$(-4x)^2 = (-4x) \times (-4x) = 16x^2\text{:}$ the brackets make the square cover the whole term.

$xy = x \times y$ (not $x + y$ ), and $xy = yx$ because multiplication can be done in either order.

To find the value of an expression, put the number in place of the letter, in brackets when it is negative, and work out the square first.

Example 2: Finding the value

Find the value of (a) $x^2 - 5x + 6$ when $x = -2$ ; (b) $-x^2$ when $x = 3$ ; (c) $(-x)^2$ when $x = 3$ .

Solution

(a) $(-2)^2 - 5(-2) + 6 = 4 + 10 + 6 = 20\text{.}$ The square of $-2$ is $(-2) \times (-2) = 4$ , and $-5 \times (-2) = +10$ .

(b) $-x^2$ means $-(x^2)$ : square first, then take the negative. $-(3^2) = -9$ .

(a) $20$ ; (b) $-9$ ; (c) $9$ . So $-x^2$ and $(-x)^2$ are not the same.

Linear Expressions

Linear expression in $x$

A linear expression in $x$ is one that can be written as $ax + b$ , where $a$ and $b$ are numbers and $a \ne 0$ .

The highest power of $x$ in it is $1$ : there is a term in $x$ , and no $x^2$ , $x^3$ , $\frac{1}{x}$ or product of letters.

For example, $7 - 4x = -4x + 7$ (here $a = -4$ , $b = 7$ ), $5x$ (here $b = 0$ ), and $\frac{2 - x}{9} = -\frac{1}{9}x + \frac{2}{9}$ are all linear in $x$ . In the same way $5y + 8$ is a linear expression in $y$ .

An expression can be linear in two variables. $4x + y - 8$ is a linear expression in $x$ and $y$ : each term is a number times one letter, or a number. It is the kind of expression in the equations $x + 2y = 7$ of chapter 2. It is not a linear expression in $x$ alone, because it also has $y$ .

A number on its own, such as $4$ , is a constant. It has no term in $x$ (it is $ax + b$ with $a = 0$ ), so it is not a linear expression.

Quadratic Expressions

Quadratic expression in $x$

A quadratic expression in $x$ is one that can be written as $ax^2 + bx + c$ , where $a$ , $b$ and $c$ are numbers and $a \ne 0$ .

The highest power of $x$ is $2$ . $a$ is the coefficient of $x^2$ , $b$ the coefficient of $x$ , and $c$ the constant term. $b$ or $c$ (or both) may be $0$ .

If $a = 0$ and $b \ne 0$ , the expression is $bx + c$ , which is linear. If $a = 0$ and $b = 0$ , it is the constant $c$ .

For example, $x^2 - 5x + 6$ ( $a = 1$ , $b = -5$ , $c = 6$ ), $3x^2 - 9x$ ( $c = 0$ ) and $-4x^2 + 3$ ( $b = 0$ ) are all quadratic. We usually write the terms in descending powers of $x$ : the $x^2$ term, then the $x$ term, then the constant.

Example 3: Linear, quadratic or neither?

State whether each expression is linear in $x$ , quadratic in $x$ , or neither, and give a reason. (a) $7 - 4x$ ; (b) $x^2 - 5x + 6$ ; (c) $4$ ; (d) $4x + y - 8$ ; (e) $x^3 + 2x$ ; (f) $\frac{1}{x} + 3$ ; (g) $6 - x^2$

Solution

(a) $7 - 4x = -4x + 7$ is $ax + b$ with $a = -4 \ne 0$ : linear in $x$ .

(b) $ax^2 + bx + c$ with $a = 1$ : quadratic in $x$ .

(d) Each term is a number times one letter, or a number, but there are two letters: linear in $x$ and $y$ , not a linear expression in $x$ alone.

(e) The highest power of $x$ is $3$ : neither.

(f) $\frac{1}{x}$ divides by $x$ ; it is not a number times $x$ or $x^2$ : neither.

(g) $6 - x^2 = -x^2 + 0x + 6\text{,}$ with $a = -1 \ne 0$ : quadratic in $x$ .

Linear in $x$ : (a). Quadratic in $x$ : (b), (g). Linear in two variables $x$ and $y$ : (d). Neither: (c), a constant; (e); (f).

Example 4: Reading off $a$ , $b$ and $c$

Write each quadratic expression in descending powers of $x$ and state the values of $a$ , $b$ and $c$ . (a) $3 - 2x^2$ ; (b) $3x^2 - 9x$ ; (c) $x - x^2 + 5$

Solution

(a) $3 - 2x^2 = -2x^2 + 3$ . There is no term in $x$ , so $b = 0$ : $a = -2$ , $b = 0$ , $c = 3$ .

(b) $3x^2 - 9x$ is already in descending powers. There is no constant term, so $c = 0$ : $a = 3$ , $b = -9$ , $c = 0$ .

(a) $-2x^2 + 3$ : $a = -2$ , $b = 0$ , $c = 3$ . (b) $a = 3$ , $b = -9$ , $c = 0$ . (c) $-x^2 + x + 5$ : $a = -1$ , $b = 1$ , $c = 5$ .

Like Terms with Powers

Like terms

Like terms have the same letters, each with the same power. Only like terms can be added or subtracted, by adding or subtracting their coefficients.

Terms	Like or unlike?	Why
$3x^2$ and $-5x^2$	like	both are a number times $x^2$
$x^2$ and $x$	unlike	$x$ has a different power in each
$xy$ and $yx$	like	$yx = xy$
$x^2y$ and $xy^2$	unlike	$x^2y = x \times x \times y$ , but $xy^2 = x \times y \times y$
$4x$ and $4$	unlike	$4$ has no letter

So $x^2 - 5x + 6$ cannot be simplified: its three terms are all unlike. And $3x^2 + 2x$ is not $5x^3$ or $5x^2$ ; it stays as it is.

Adding, Subtracting and Simplifying

To simplify, remove any brackets, collect the like terms together, and add or subtract their coefficients. Write the answer in descending powers.

Example 5: Collecting like terms

Simplify (a) $5x^2 + (-11x^2)$ ; (b) $-3x^2 + (-4x^2) + 2 - 8\text{.}$

Solution

(a) Adding $-11x^2$ is the same as subtracting $11x^2$ : $5x^2 - 11x^2 = (5 - 11)x^2 = -6x^2\text{.}$

(b) $-3x^2 - 4x^2 + 2 - 8$ . The $x^2$ terms: $-3 - 4 = -7$ . The numbers: $2 - 8 = -6$ .

$= -7x^2 - 6$

(a) $-6x^2$ ; (b) $-7x^2 - 6$ .

Example 6: Terms in $xy$

Simplify $-4y^2 - yx + 3y^2 - (-5xy)\text{.}$

Solution

Subtracting $-5xy$ is adding $5xy$ , and $yx = xy$ , so $-yx = -xy$ :

$-4y^2 - xy + 3y^2 + 5xy$

The $y^2$ terms: $-4y^2 + 3y^2 = -y^2$ . The $xy$ terms: $-xy + 5xy = 4xy$ .

$= -y^2 + 4xy$

$-y^2 + 4xy$

To subtract a whole expression, put it in brackets. The minus sign in front changes the sign of every term inside: $-(x^2 + 4x - 5) = -x^2 - 4x + 5\text{.}$

Example 7: Subtracting a quadratic expression

Simplify $(3x^2 - 2x + 1) - (x^2 + 4x - 5)\text{.}$

Solution

Remove the brackets; the minus changes every sign in the second: $3x^2 - 2x + 1 - x^2 - 4x + 5$

$x^2$ terms: $3x^2 - x^2 = 2x^2$ . $x$ terms: $-2x - 4x = -6x$ . Numbers: $1 + 5 = 6$ .

$= 2x^2 - 6x + 6$

$2x^2 - 6x + 6$

Example 8: The order of the words

Subtract $2x^2 - x$ from $5x^2 + 3x$ .

Solution

"Subtract A from B" means $B - A$ : we start with $5x^2 + 3x$ and take away $2x^2 - x$ .

$(5x^2 + 3x) - (2x^2 - x) = 5x^2 + 3x - 2x^2 + x$

$= 3x^2 + 4x$

$3x^2 + 4x$

Writing an Expression for an Area

Quadratic expressions arise naturally from areas: a length in $x$ times another length in $x$ gives a term in $x^2$ .

Example 9: A square and a rectangle

A square of side $x$ cm is placed next to a rectangle $x$ cm by $3$ cm, as shown (lengths in cm). Write down an expression for (a) the total area, (b) the perimeter of the whole shape. Say which is quadratic and which is linear.

Solution

(a) Area of the square $= x \times x = x^2$ cm². Area of the rectangle $= 3 \times x = 3x$ cm².

Total area $= (x^2 + 3x)$ cm².

(b) The whole shape is a rectangle $(x + 3)$ cm long and $x$ cm high. Perimeter $= (x + 3) + x + (x + 3) + x = (4x + 6)$ cm.

$x^2 + 3x$ has highest power 2, so it is quadratic. $4x + 6$ is linear.

The area is $(x^2 + 3x)$ cm², a quadratic expression; the perimeter is $(4x + 6)$ cm, a linear expression.

Common Mistakes

Reading $x^2$ as $2x$ . $x^2 = x \times x$ ; when $x = 5$ , $x^2 = 25$ but $2x = 10$ .
Squaring the sign in $-x^2$ . When $x = 3$ , $-x^2 = -9$ , not $9$ ; only $(-x)^2$ is $9$ .
Adding unlike terms: $3x^2 + 2x$ is not $5x^3$ or $5x^2$ . It cannot be simplified.
Calling a constant such as $4$ a linear expression. It has no term in $x$ .
Calling $4x + y - 8$ "not linear". It is linear in two variables, $x$ and $y$ .
Changing only the first sign when a bracket is subtracted: $-(x^2 + 4x - 5)$ is $-x^2 - 4x + 5$ , not $-x^2 + 4x - 5$ .
Writing "subtract $A$ from $B$ " as $A - B$ . It is $B - A$ .

Practice

Try each question before you open its solution.

Question 1

For $7x^2 - x + 4 - 2xy$ , write down the coefficient of $x^2$ , the coefficient of $x$ , the coefficient of $xy$ , and the constant term.

ShowHide solution

The terms, each with its sign, are $7x^2$ , $-x$ , $4$ and $-2xy$ .

$-x = -1 \times x$ , so its coefficient is $-1$ .

Coefficient of $x^2$ : $7$ ; of $x$ : $-1$ ; of $xy$ : $-2$ ; constant term: $4$ .

Question 2

State whether each is linear in $x$ , quadratic in $x$ , or neither, with a reason. (a) $9 - 2x$ ; (b) $x^2 + 1$ ; (c) $-6$ ; (d) $\frac{6 - x}{2}$ ; (e) $5x^3 - x$ ; (f) $3a + 2b - 1$

ShowHide solution

(a) $-2x + 9$ : $ax + b$ with $a = -2$ . Linear.

(b) $x^2 + 0x + 1$ : $a = 1$ . Quadratic.

(d) $\frac{6 - x}{2} = -\frac{1}{2}x + 3\text{:}$ $a = -\frac{1}{2}$ . Linear.

(e) Highest power of $x$ is $3$ . Neither.

(f) A number times one letter in each term, but with two letters $a$ and $b$ and no $x$ : it is linear in $a$ and $b$ , not an expression in $x$ . Neither.

Linear in $x$ : (a), (d). Quadratic in $x$ : (b). Neither: (c), (e), and (f), which has no $x$ ; (f) is linear in the two variables $a$ and $b$ .

Question 3

Write each in descending powers of $x$ and state $a$ , $b$ and $c$ . (a) $4 - x^2$ ; (b) $2x - 5x^2$ ; (c) $7 + 3x + x^2$

ShowHide solution

(a) $-x^2 + 4$ : no $x$ term, so $b = 0$ .

(b) $-5x^2 + 2x$ : no constant term, so $c = 0$ .

(a) $a = -1$ , $b = 0$ , $c = 4$ ; (b) $a = -5$ , $b = 2$ , $c = 0$ ; (c) $a = 1$ , $b = 3$ , $c = 7$ .

Question 4

Find the value of each expression when $x = -3$ . (a) $2x^2 - x + 1$ ; (b) $-x^2 + 4x$ ; (c) $(2x)^2$ and $2x^2$

ShowHide solution

(a) $2(-3)^2 - (-3) + 1 = 2(9) + 3 + 1 = 22\text{.}$

(b) $-(-3)^2 + 4(-3) = -9 - 12 = -21\text{.}$

(a) $22$ ; (b) $-21$ ; (c) $(2x)^2 = 36$ and $2x^2 = 18$ : they are different.

Question 5

Are these pairs like or unlike terms? (a) $3x^2$ and $-x^2$ ; (b) $4x$ and $4x^2$ ; (c) $2ab$ and $-5ba$ ; (d) $a^2b$ and $ab^2$

ShowHide solution

(a) Both are a number times $x^2$ .

(b) $x$ and $x^2$ are different powers.

(d) $a^2b = a \times a \times b$ and $ab^2 = a \times b \times b$ .

(a) like; (b) unlike; (c) like; (d) unlike.

Question 6

Simplify (a) $8x^2 - 15x^2$ ; (b) $-2y^2 + 9y^2 - 7y^2$ ; (c) $6x^2 - 4x + 3 - 2x^2 + 9x - 8$

ShowHide solution

(a) $(8 - 15)x^2 = -7x^2$ .

(b) $(-2 + 9 - 7)y^2 = 0y^2 = 0\text{.}$

(a) $-7x^2$ ; (b) $0$ ; (c) $4x^2 + 5x - 5$ .

Question 7

Simplify (a) $3ab - 5ba + 2a^2$ ; (b) $x^2 - 3xy + 5yx - 4x^2$

ShowHide solution

(a) $ba = ab$ : $3ab - 5ab = -2ab$ . The answer in descending powers of $a$ is $2a^2 - 2ab$ .

(b) $x^2 - 4x^2 = -3x^2$ , and $-3xy + 5xy = 2xy$ .

(a) $2a^2 - 2ab$ ; (b) $-3x^2 + 2xy$ .

Question 8

Simplify (a) $(4x^2 - 3x + 2) - (x^2 - 5x + 7)\text{;}$ (b) $(2x^2 + x - 1) + (-5x^2 + 4x + 6)$

ShowHide solution

(a) $4x^2 - 3x + 2 - x^2 + 5x - 7\text{:}$ $3x^2 + 2x - 5$ .

(b) The plus sign leaves the signs as they are: $2x^2 + x - 1 - 5x^2 + 4x + 6\text{:}$ $-3x^2 + 5x + 5$ .

(a) $3x^2 + 2x - 5$ ; (b) $-3x^2 + 5x + 5$ .

Question 9

Subtract $3x^2 - 2x + 4$ from $x^2 + x$ .

ShowHide solution

Start with $x^2 + x$ and take away $3x^2 - 2x + 4$ :

$(x^2 + x) - (3x^2 - 2x + 4) = x^2 + x - 3x^2 + 2x - 4$

$= -2x^2 + 3x - 4$

$-2x^2 + 3x - 4$

Question 10

A square photograph of side $x$ cm is stuck on a square card of side $3x$ cm. Write down an expression for the area of the card that is not covered.

ShowHide solution

Area of the card $= (3x)^2 = 3x \times 3x = 9x^2$ cm².

Area of the photograph $= x^2$ cm².

Area not covered $= 9x^2 - x^2 = 8x^2$ cm².

The uncovered area is $8x^2$ cm².

Question 11

A square room of side $x$ m has a corridor $x$ m long and $2$ m wide along one side, so that together they make a rectangle $x$ m by $(x + 2)$ m. (a) Write the total floor area as the sum of the two areas. (b) Write the perimeter of the whole rectangle. (c) Which expression is quadratic and which is linear? (d) Find both when $x = 5$ .

ShowHide solution

(a) Room: $x \times x = x^2$ . Corridor: $x \times 2 = 2x$ . Total: $(x^2 + 2x)$ m².

(b) $x + (x + 2) + x + (x + 2) = (4x + 4)$ m.

(d) Area: $5^2 + 2(5) = 25 + 10 = 35\text{.}$ Perimeter: $4(5) + 4 = 24$ . Check: the rectangle is $5$ m by $7$ m, area $35$ m², perimeter $5 + 7 + 5 + 7 = 24$ m.

(a) $(x^2 + 2x)$ m²; (b) $(4x + 4)$ m; (c) the area is quadratic, the perimeter linear; (d) $35$ m² and $24$ m.

Question 12

Wei Jie made one mistake in each line. Find it and correct it. (a) $5x^2 - 2x = 3x$ ; (b) $-(x^2 - 4x) = -x^2 - 4x\text{;}$ (c) when $x = -2$ , $-x^2 = 4$

ShowHide solution

(a) $5x^2$ and $2x$ are unlike terms, so one cannot be taken from the other. $5x^2 - 2x$ stays as it is.

(b) The minus changes the sign of every term: $-(x^2 - 4x) = -x^2 + 4x\text{.}$

(a) $5x^2 - 2x$ cannot be simplified; (b) $-x^2 + 4x$ ; (c) $-4$ .

Key Points

Each term takes the sign in front of it; a coefficient is the number multiplying the letters, sign included.
$x^2 = x \times x$ , not $2x$ ; $-4x^2 = -4 \times x^2$ , while $(-4x)^2 = 16x^2$ .
A linear expression in $x$ can be written $ax + b$ with $a \ne 0$ ; a constant is not linear.
$4x + y - 8$ is linear in two variables, $x$ and $y$ .
A quadratic expression in $x$ can be written $ax^2 + bx + c$ with $a \ne 0$ ; $b$ and $c$ may be $0$ .
Like terms have the same letters with the same powers; only like terms can be collected.
A minus sign in front of a bracket changes the sign of every term inside; "subtract $A$ from $B$ " is $B - A$ .

Quadratic Expressions

Before you start

Terms, Coefficients and Constant Terms

Coefficients and the constant term

Example 1: Naming the parts

What x2x^2x2, −4x2-4x^2−4x2 and xyxyxy Mean

Notation

Example 2: Finding the value

Linear Expressions

Linear expression in xxx

Quadratic Expressions

Quadratic expression in xxx

Example 3: Linear, quadratic or neither?

Example 4: Reading off aaa, bbb and ccc

Like Terms with Powers

Like terms

Adding, Subtracting and Simplifying

Example 5: Collecting like terms

Example 6: Terms in xyxyxy

Example 7: Subtracting a quadratic expression

Example 8: The order of the words

Writing an Expression for an Area

Example 9: A square and a rectangle

Common Mistakes

Practice

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Key Points

What $x^2$ , $-4x^2$ and $xy$ Mean

Linear expression in $x$

Quadratic expression in $x$

Example 4: Reading off $a$ , $b$ and $c$

Example 6: Terms in $xy$