MPM2D · Unit 5 test review

Quadratic Equations: Test Review

Everything for the Unit 5 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

Solving Quadratic Equations by Factoring

Zero product property: if $A \times B = 0$ , then $A = 0$ or $B = 0$ . It works only when the product equals 0.
Standard form of a quadratic equation: $a x^{2} + b x + c = 0$ with $a \neq = 0$ . Rearrange to this form before factoring.
The roots of $a x^{2} + b x + c = 0$ are the zeros ( $x$ -intercepts) of $y = a x^{2} + b x + c$ .
If $a (x - r) (x - s) = 0$ with $a \neq = 0$ , the roots are $x = r$ and $x = s$ . A factor $(x + 2)$ gives the root $- 2$ .
A perfect square such as $(x - 3)^{2} = 0$ has one distinct root (a double root), where the vertex touches the $x$ -axis.
Verify each root by substituting it into the original equation; in context, reject roots that make no sense.

The Quadratic Formula

Quadratic formula: for $a x^{2} + b x + c = 0$ with $a \neq = 0$ , $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ .
The formula comes from completing the square on $a x^{2} + b x + c = 0$ .
The equation must be in standard form before you read off $a$ , $b$ and $c$ , with their signs.
The fraction bar is under the whole numerator $- b \pm b^{2} - 4 a c$ .
The roots are equally spaced on either side of $x = - \frac{b}{2 a}$ , the axis of symmetry.
If $a$ , $b$ , $c$ are integers and $b^{2} - 4 a c$ is a perfect square, the roots are rational and the equation can be factored.

Real and Non-Real Roots

The discriminant of $a x^{2} + b x + c = 0$ is $b^{2} - 4 a c$ , the expression under the square root in the quadratic formula.
$b^{2} - 4 a c > 0$ : two distinct real roots; the parabola has two $x$ -intercepts.
$b^{2} - 4 a c = 0$ : one real root, $x = - \frac{b}{2 a}$ ; the vertex is on the $x$ -axis.
$b^{2} - 4 a c < 0$ : no real roots; the parabola does not meet the $x$ -axis. (The roots are non-real.)
From $y = a (x - h)^{2} + k$ : if $a$ and $k$ have opposite signs there are 2 zeros, if they have the same sign there are none, and if $k = 0$ there is 1.
With integer $a$ , $b$ , $c$ : a perfect-square discriminant means rational roots, so the equation factors.

Solving Problems with Quadratics

Vertical motion: $h = - 4.9 t^{2} + v_{0} t + h_{0}$ ( $h$ in metres, $t$ in seconds, air resistance ignored).
The maximum or minimum value is the second coordinate of the vertex; it occurs at $x = - \frac{b}{2 a}$ , which is halfway between the zeros when the parabola has two.
"When does it reach height $H$ ?" means solve $h = H$ . Reject roots outside the domain, such as negative times.
For a parabola that opens down, the height is above $H$ between the two roots of $h = H$ .
For maximum area or revenue, write the quantity as a product of two factors, find the zeros, and go halfway between them.

Where marks get lost

Using the zero product property when the right side is not 0: from $x (x - 5) = 6$ , writing $x = 6$ or $x - 5 = 6$ is wrong. First rearrange to $x^{2} - 5 x - 6 = 0$ , then factor as $(x - 6) (x + 1) = 0$ to get $x = 6$ or $x = - 1$ . (Solving Quadratic Equations by Factoring)
Keeping the sign from the bracket: $(x + 2) (x + 8) = 0$ gives $x = - 2$ and $x = - 8$ , not $x = 2$ and $x = 8$ . Ask which number makes each bracket equal 0. (Solving Quadratic Equations by Factoring)
Dividing both sides by $x$ : from $x^{2} = 5 x$ , dividing gives only $x = 5$ and loses the root $x = 0$ . Factor $x^{2} - 5 x = x (x - 5)$ instead. (Solving Quadratic Equations by Factoring)
Solving $2 x + 1 = 0$ as $x = - 2$ or $x = \frac{1}{2}$ . Subtract 1 first, then divide by 2: $x = - \frac{1}{2}$ . (Solving Quadratic Equations by Factoring)
Giving one root for $x^{2} = 25$ . Both $5^{2}$ and $(- 5)^{2}$ equal 25, so $x = 5$ or $x = - 5$ . (Solving Quadratic Equations by Factoring)
Forgetting that $- b$ changes the sign of $b$ : for $b = - 5$ , $- b = + 5$ , not $- 5$ . (The Quadratic Formula)
Typing $- 5^{2}$ on a calculator for $b^{2}$ and getting $- 25$ . Any non-zero number squared is positive: $(- 5)^{2} = 25$ . (The Quadratic Formula)
Dividing only the square root by $2 a$ , as in $x = - 1 \pm \frac{17}{2}$ . The fraction bar goes under the whole numerator: $x = \frac{- 1 \pm 17}{2}$ . (The Quadratic Formula)
Reading $a$ , $b$ , $c$ before rearranging: in $2 x^{2} = 3 - 4 x$ , the values are not $b = - 4$ , $c = 3$ . First rewrite it as $2 x^{2} + 4 x - 3 = 0$ , so $b = 4$ and $c = - 3$ . (The Quadratic Formula)
Rounding the square root early. Using $17 \approx 4$ gives $x = 1.5$ instead of $1.56$ ; keep the calculator value until the end. (The Quadratic Formula)
Sign errors in $- 4 a c$ : for $- x^{2} + 2 x + 3 = 0$ , $b^{2} - 4 a c = 4 - 4 (- 1) (3) = 4 + 12 = 16,$ not $4 - 12 = - 8$ . Write the signs of $a$ and $c$ in brackets. (Real and Non-Real Roots)
Thinking a discriminant of 0 means no roots. A discriminant of 0 means one real root, $x = - \frac{b}{2 a}$ ; only a negative discriminant means no real roots. (Real and Non-Real Roots)
Calculating the discriminant before rearranging: for $x^{2} + 5 = 2 x$ , first write $x^{2} - 2 x + 5 = 0$ , so $b^{2} - 4 a c = 4 - 20 = - 16$ . (Real and Non-Real Roots)
Assuming a vertex above the $x$ -axis means no zeros. That is true only if the parabola opens up. $y = - 2 (x - 3)^{2} + 5$ has its vertex above the axis, opens down, and has two zeros. (Real and Non-Real Roots)
Writing "no solution" without saying what kind. The precise statement is "no real roots": the equation has non-real roots, which you will study in later courses. (Real and Non-Real Roots)
Giving the time of the maximum as the maximum height. For $h = - 4.9 t^{2} + 14 t + 1.5$ , the vertex is about $(1.43, 11.5)$ : the ball is highest at 1.43 s, and the maximum height is 11.5 m. (Solving Problems with Quadratics)
Keeping a negative time. When $h = 0$ gives $t \approx - 0.10$ or $t \approx 2.96$ , the negative root is before the throw and must be rejected. (Solving Problems with Quadratics)
Giving the wrong part of the time line. For a parabola that opens down, "above 8 m" is the interval between the two roots, not before the first or after the second. (Solving Problems with Quadratics)
Sign slips in $t = - \frac{b}{2 a}$ : with $b = 50$ and $a = - 4.9$ , $- \frac{50}{2 ( - 4.9 )} = + 5.10$ , not $- 5.10$ . (Solving Problems with Quadratics)
Ignoring the wall in a fencing problem: with a wall on one side, the fence covers $2 x + length$ , not $2 x + 2 \times length$ . (Solving Problems with Quadratics)

Practice test

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

Solving Quadratic Equations by Factoring
1
Solve $x^{2} - 7 x + 12 = 0$ .
Solving Quadratic Equations by Factoring
2
Solve $2 x^{2} - 7 x + 3 = 0$ . Enter the smaller root as a decimal or a fraction.
Solving Quadratic Equations by Factoring
3
Put the steps for solving $x^{2} = 2 x + 8$ in order.
1. 1Verify: $16 = 2 (4) + 8$ and $4 = 2 (- 2) + 8$
2. 2Solve: $x = 4$ or $x = - 2$
3. 3Set each factor to zero: $x - 4 = 0$ or $x + 2 = 0$
4. 4Subtract $2 x$ and $8$ from both sides: $x^{2} - 2 x - 8 = 0$
5. 5Factor: $(x - 4) (x + 2) = 0$
Solving Quadratic Equations by Factoring
4
Where does the graph of $y = (x - 3) (x + 5)$ cross the $x$ -axis?
Solving Quadratic Equations by Factoring
5
How many different roots does $x^{2} - 10 x + 25 = 0$ have?
Solving Quadratic Equations by Factoring
6
Name the property that says: if a product of factors equals 0, then at least one factor equals 0.
The Quadratic Formula
7
To use the quadratic formula on $3 x^{2} - x = 7$ , what are $a$ , $b$ and $c$ ?
The Quadratic Formula
8
Solve $x^{2} + x - 4 = 0$ . Enter the positive root, rounded to two decimal places.
The Quadratic Formula
9
Which is the correct substitution into the quadratic formula for $2 x^{2} - 5 x + 1 = 0$ ?
The Quadratic Formula
10
Match each part of the quadratic formula with what it does.
$b^{2} - 4 a c$
$\pm$
$- \frac{b}{2 a}$
$2 a$
The Quadratic Formula
11
Solve $4 x^{2} + 4 x - 3 = 0$ using the quadratic formula. Enter the larger root.
The Quadratic Formula
12
In the quadratic formula, what expression is under the square root sign? Type powers with ^: type b^2 for $b^{2}$ .
Real and Non-Real Roots
13
Calculate the discriminant of $3 x^{2} - 5 x + 2 = 0$ .
Real and Non-Real Roots
14
The graph of $y = a x^{2} + b x + c$ touches the $x$ -axis at exactly one point. What must be true?
Real and Non-Real Roots
15
How many zeros does $y = - 2 (x - 3)^{2} + 5$ have?
Real and Non-Real Roots
16
What is the largest integer $k$ for which $x^{2} + 4 x + k = 0$ has two real roots?
Real and Non-Real Roots
17
Put these equations in order from the smallest discriminant to the largest.
1. 1 $x^{2} + 2 x + 1 = 0$
2. 2 $x^{2} + 5 x = 0$
3. 3 $x^{2} - 1 = 0$
4. 4 $x^{2} + 4 = 0$
Real and Non-Real Roots
18
Which statement about $x^{2} - 4 x + 5 = 0$ is correct?
Solving Problems with Quadratics
19
A ball's height is $h = - 4.9 t^{2} + 14 t + 1.5$ metres after $t$ seconds. What is its maximum height?
m
Solving Problems with Quadratics
20
A projectile's height is $h = - 4.9 t^{2} + 50 t + 1.5$ metres after $t$ seconds. Find its maximum height, rounded to one decimal place.
m
Solving Problems with Quadratics
21
For that same ball, for how many seconds is it higher than 8 m? Round to the nearest hundredth.
s
Solving Problems with Quadratics
22
What is the minimum value of $y = 2 x^{2} - 12 x + 23$ ?
Solving Problems with Quadratics
23
For a thrown ball modelled by $h = - 4.9 t^{2} + v_{0} t + h_{0}$ , match each question with the feature of the graph that answers it.
What is the maximum height?
When is the ball highest?
When does the ball land?
How high was the ball when it was thrown?
Solving Problems with Quadratics
24
In the model $h = - 4.9 t^{2} + v_{0} t + h_{0}$ , what does $h_{0}$ represent? Answer in a few words.

Endless drill

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

Endless drillnew numbers every time

Generating a problem…