MPM2D · Unit 3 test review

Quadratic Expressions: Test Review

Everything for the Unit 3 test on one page: the key facts from all 4 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Expanding and Simplifying

Distributive property: $a (b + c) = ab + a c$ . Every term outside multiplies every term inside.
Binomial times binomial gives four products before collecting like terms: $(a + b) (c + d) = a c + a d + b c + b d$ .
$(x + m) (x + n)$ expands to $x^{2} + (m + n) x + mn$ : the middle coefficient is the sum and the constant is the product.
Squares of binomials: $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ and $(a - b)^{2} = a^{2} - 2 ab + b^{2}$ .
Product of a sum and a difference: $(a + b) (a - b) = a^{2} - b^{2}$ .
Check an expansion by substituting the same value of $x$ into the original and the answer.

Common Factors and Grouping

GCF of monomials: the GCF of the coefficients times each shared variable to its lowest power.
Common factoring: $ab + a c = a (b + c)$ . Divide every term by the GCF; a term equal to the GCF leaves 1.
A common binomial factor comes out like a single term: $3 x (x - 2) + 5 (x - 2) = (x - 2) (3 x + 5) .$
Grouping: $a x + a y + b x + b y = a (x + y) + b (x + y) = (x + y) (a + b) .$
Opposite brackets: $b - a = - (a - b)$ .
Always check a factorization by expanding it back.

Factoring Trinomials

$x^{2} + b x + c = (x + m) (x + n)$ when $m + n = b$ and $mn = c$ .
Positive $c$ : same signs (the sign of $b$ ). Negative $c$ : opposite signs, and the number farther from zero has the sign of $b$ .
Decomposition for $a x^{2} + b x + c$ : find two numbers with product $a c$ and sum $b$ , split the middle term, then factor by grouping.
Take out any common factor first, including $- 1$ when the $x^{2}$ term is negative.
If no integer pair has the right product and sum, the trinomial does not factor using integers.
Check every factorization by expanding.

Perfect Squares and Difference of Squares

Difference of squares: $a^{2} - b^{2} = (a - b) (a + b)$ .
Perfect square trinomials: $a^{2} + 2 ab + b^{2} = (a + b)^{2}$ and $a^{2} - 2 ab + b^{2} = (a - b)^{2}$ .
Perfect square test: first and last terms are squares, the last term is positive, and the middle term is $\pm 2 ab$ .
A second-degree sum of squares such as $x^{2} + 25$ has no binomial factors using real numbers.
Factoring strategy: common factor first, count the terms, factor again where possible, then check by expanding.

Where marks get lost

Squaring each term separately: $(x + 5)^{2} = x^{2} + 25$ is wrong. The square of a binomial has a middle term: $(x + 5)^{2} = x^{2} + 10 x + 25$ . (Expanding and Simplifying)
Losing a negative sign: in $(x - 4) (x + 7)$ the last product is $- 4 \times 7 = - 28$ , not $+ 28$ . (Expanding and Simplifying)
Subtracting only the first term of a bracket: $- (x - 4)^{2}$ is $- (x^{2} - 8 x + 16) = - x^{2} + 8 x - 16$ , not $- x^{2} - 8 x + 16$ . (Expanding and Simplifying)
Adding unlike terms: $2 x^{2} + 5 x y$ cannot be combined into $7 x^{3} y$ or anything else, because the variable parts are different. (Expanding and Simplifying)
Distributing a number in front into both brackets: $2 (x + 3) (x - 1)$ is not $(2 x + 6) (2 x - 2)$ , which is 4 times the product. (Expanding and Simplifying)
Leaving out the 1: $6 x^{3} - 3 x^{2} + 3 x = 3 x (2 x^{2} - x)$ is wrong. The last term divided by $3 x$ is 1, so the answer is $3 x (2 x^{2} - x + 1)$ . (Common Factors and Grouping)
Not taking out the greatest factor: $12 x^{2} - 8 x = 2 x (6 x - 4)$ is equal but not fully factored, because $6 x - 4$ still has a factor 2. The full answer is $4 x (3 x - 2)$ . (Common Factors and Grouping)
Forgetting to change signs when taking out a negative: $- 4 x^{2} + 8 x = - 4 x (x + 2)$ is wrong; $8 x \div (- 4 x) = - 2$ , so it is $- 4 x (x - 2)$ . (Common Factors and Grouping)
Grouping with the wrong sign on the second pair: $x^{2} + 3 x - 2 x - 6 = x (x + 3) + 2 (- x - 3)$ leaves brackets that do not match. Take out $- 2$ to get $- 2 (x + 3)$ . (Common Factors and Grouping)
Stopping after the first grouping step: $2 (x - y) + a (x - y)$ is still a sum, not a product. Take out the common bracket to finish. (Common Factors and Grouping)
Matching only the product: for $x^{2} + 5 x + 6$ , choosing 1 and 6 (product 6, but sum 7). Both conditions must hold, so the pair is 2 and 3. (Factoring Trinomials)
Getting the signs backwards: $x^{2} - x - 6 = (x - 2) (x + 3)$ is wrong; it expands to $x^{2} + x - 6$ . Check the middle term every time. (Factoring Trinomials)
Forgetting the common factor: $3 x^{2} - 15 x + 18$ factored as $(3 x - 6) (x - 3)$ is equal but not fully factored, because $3 x - 6 = 3 (x - 2)$ . (Factoring Trinomials)
Using the product $c$ instead of $a c$ when $a \neq = 1$ : for $2 x^{2} + 11 x + 5$ you need product $10$ , not $5$ . (Factoring Trinomials)
Dropping the $- 1$ when grouping: $6 x^{2} + 9 x - 2 x - 3 = 3 x (2 x + 3) - 2 x - 3$ is unfinished. Write $- 2 x - 3 = - 1 (2 x + 3)$ . (Factoring Trinomials)
Factoring a sum of squares as if it were a difference: $x^{2} + 25 \neq = (x + 5) (x - 5)$ , because that product is $x^{2} - 25$ . (Perfect Squares and Difference of Squares)
Calling every trinomial with square first and last terms a perfect square: $x^{2} + 10 x + 16$ has $16 = 4^{2}$ , but its middle term would need to be $8 x$ . (Perfect Squares and Difference of Squares)
Taking the square root of only the coefficient or only the variable: $16 x^{2} = (4 x)^{2}$ , not $(16 x)^{2}$ or $(8 x)^{2}$ . (Perfect Squares and Difference of Squares)
Stopping too early: $2 x^{2} - 18 = 2 (x^{2} - 9)$ is not fully factored; the bracket is a difference of squares, so the answer is $2 (x - 3) (x + 3)$ . (Perfect Squares and Difference of Squares)
Missing a common factor that hides the pattern: $3 x^{2} - 48$ is not a difference of squares as written, but $3 (x^{2} - 16)$ contains one. (Perfect Squares and Difference of Squares)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Expanding and Simplifying
1
Expand and simplify $(x + 6) (x - 2)$ .
Expanding and Simplifying
2
What is the constant term of $(2 x - 5) (3 x - 7)$ after expanding?
Expanding and Simplifying
3
Expand and simplify $(x - 6)^{2}$ . Type powers with ^: type x^2 for $x^{2}$ .
Expanding and Simplifying
4
Match each product with its expanded form.
1. A. $x^{2} - 2 x - 3$
2. B. $x^{2} - 6 x + 9$
3. C. $x^{2} - 9$
4. D. $x^{2} + 6 x + 9$
$(x + 3) (x - 3)$
$(x - 3)^{2}$
$(x + 3)^{2}$
$(x - 3) (x + 1)$
Expanding and Simplifying
5
Simplify $(x + 2) (x + 5) - (x + 1) (x + 3)$ .
Expanding and Simplifying
6
Expand $3 x (2 x - 5)$ . Type powers with ^: type x^2 for $x^{2}$ .
Common Factors and Grouping
7
What is the greatest common factor of $24 x^{2} y$ and $36 x y^{2}$ ?
Common Factors and Grouping
8
Factor fully: $6 x^{2} - 9 x$ .
Common Factors and Grouping
9
$18 x^{2} y - 24 x y^{2}$ factors fully as $k x y (3 x - 4 y)$ . What is $k$ ?
Common Factors and Grouping
10
Factor: $3 x (x - 2) + 5 (x - 2)$ .
Common Factors and Grouping
11
Use a common factor to calculate $17 \times 23 + 17 \times 77$ without a calculator.
Common Factors and Grouping
12
Factor: $5 x (x - 3) - 2 (3 - x)$ .
Factoring Trinomials
13
Factor $x^{2} + 10 x + 16$ .
Factoring Trinomials
14
The trinomial $x^{2} + b x - 24$ factors as $(x + 6) (x - 4)$ . What is $b$ ?
Factoring Trinomials
15
Which trinomial cannot be factored using integers?
Factoring Trinomials
16
Put the steps for factoring $2 x^{2} + 11 x + 5$ by decomposition in order.
1. 1Take out the common bracket: $(x + 5) (2 x + 1)$
2. 2Common factor each pair: $2 x (x + 5) + 1 (x + 5)$
3. 3Split the middle term: $2 x^{2} + 10 x + x + 5$
4. 4Calculate $a c = 2 \times 5 = 10$
5. 5Find the pair with product 10 and sum 11: 10 and 1
Factoring Trinomials
17
For how many integer values of $k$ can $x^{2} + k x + 12$ be factored as $(x + m) (x + n)$ with integers $m$ and $n$ ?
Factoring Trinomials
18
Match each trinomial with its factored form.
1. A. $(x - 3) (x - 4)$
2. B. $(x - 4) (x + 3)$
3. C. $(x + 3) (x + 4)$
4. D. $(x + 4) (x - 3)$
$x^{2} + 7 x + 12$
$x^{2} - 7 x + 12$
$x^{2} + x - 12$
$x^{2} - x - 12$
Perfect Squares and Difference of Squares
19
Factor $x^{2} - 49$ .
Perfect Squares and Difference of Squares
20
Use a difference of squares to calculate $49 \times 51$ without a calculator.
Perfect Squares and Difference of Squares
21
Which expression cannot be factored using real numbers?
Perfect Squares and Difference of Squares
22
Match each expression with the best description of how it factors.
$x^{2} - 64$
$x^{2} - 16 x + 64$
$x^{2} + 64$
$x^{2} + 16 x$
Perfect Squares and Difference of Squares
23
Which is a perfect square trinomial?
Perfect Squares and Difference of Squares
24
Factor $(x + 3)^{2} - 16$ .

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…