VertexMath · MPM2D · Free worksheet

Solving Quadratic Equations Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Zero product property: if $ab=0$ , then $a=0$ or $b=0$ . It works only when one side is $0$ , so move every term to one side first.
Completing the square: for $x^2+bx$ , add and subtract $\left(\frac{b}{2}\right)^2$ . If $a\ne1$ , factor $a$ out of the $x$ -terms first: what you add inside the bracket is multiplied by $a$ .
Quadratic formula: for $ax^2+bx+c=0$ , $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ . The $2a$ divides the whole numerator.
Discriminant $D=b^2-4ac$ : $D>0$ gives two real roots, $D=0$ one, $D<0$ none.
In a word problem, test every root against the situation. A root can be a negative length, a width bigger than the space, or a moment after the model stops applying.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Factoring · Completing the Square · Quadratic Formula · Discriminant

K1 Zero Product Property

Solve each equation.

a)
$(x-4)(3x+2)=0$
b)
$x^2=6x$
c)
$(x-3)(x+1)=5$

Stuck? Peek at a hint

Get $0$ alone on one side before you use the zero product property, then factor.

K2 Roots on the Graph

The graph of $y=-x^2+2x+8$ is shown. Each grid square is 1 unit.

a)
Use the graph to solve $-x^2+2x+8=0$ .
b)
Use the graph to solve $-x^2+2x+8=5$ .
c)
Determine the number of solutions of $-x^2+2x+8=k$ when $k=9$ and when $k=10$ .

Stuck? Peek at a hint

The solutions of $-x^2+2x+8=k$ are the $x$ -coordinates of the points where the graph meets the horizontal line $y=k$ .

K3 Completing the Square

Solve by completing the square. Give exact answers, or state that there are no real roots.

a)
$x^2+10x+18=0$
b)
$x^2-6x+14=0$
c)
$2x^2-12x+7=0$

Stuck? Peek at a hint

Move the constant to the right side, then add $\left(\frac{b}{2}\right)^2$ to both sides; in (c), make the coefficient of $x^2$ equal to $1$ first.

K4 Quadratic Formula

Solve using the quadratic formula. Give exact answers in simplest form.

a)
$x^2-4x-6=0$
b)
$3x^2+2x-2=0$
c)
$2x^2=5x+1$ . Also give the roots to two decimal places.

Stuck? Peek at a hint

Write each equation as $ax^2+bx+c=0$ and list $a$ , $b$ and $c$ with their signs before you substitute.

K5 The Discriminant

Without solving, find the value of the discriminant and state the number of real roots.

a)
$x^2-6x+9=0$
b)
$3x^2-4x+5=0$
c)
$-2x^2+3x+4=0$

Stuck? Peek at a hint

Compute $b^2-4ac$ with brackets around every negative value, then look only at its sign.

Real World (A)

Projectiles · Area · Profit

A1 Gym Ceiling

In a school gym in Sudbury, a student bumps a volleyball upward. Its height, in metres, after $t$ seconds is modelled by $h(t)=-5t^2+12t+2$ .

a)
If nothing were in the way, when would the ball hit the floor? Round to the nearest hundredth of a second.
b)
Use the discriminant to determine whether the ball reaches a height of $10$ m.
c)
The gym ceiling is $8$ m high. Solve $h(t)=8$ . What does this mean for your answer to (a)?

Stuck? Peek at a hint

Hitting the floor means $h(t)=0$ ; reaching a height $H$ means $h(t)=H$ has a real solution.

A2 Garden Path

A rectangular community garden in Guelph measures $20$ m by $12$ m. A gravel path of uniform width $x$ metres runs along all four sides, inside the fence, as shown (not to scale). The planting bed in the middle must have an area of $128$ m².

a)
Write an equation for the area of the planting bed and rearrange it to the form $ax^2+bx+c=0$ .
b)
Solve the equation.
c)
Which solution fits the garden? State the path width and the dimensions of the planting bed.

Stuck? Peek at a hint

The path takes $x$ metres off both ends of each side, so the bed is $(20-2x)$ m long.

A3 Mug Market

A student sells handmade mugs at a weekend market in Kingston. Each mug costs $10 to make. The table shows how many mugs, $n$ , sell per day at a price of $x$ dollars per mug.

$x$	$15$	$20$	$25$	$30$
$n$	$125$	$100$	$75$	$50$

a)
Write an equation for $n$ in terms of $x$ . Then show that the daily profit, in dollars, is $P(x)=-5x^2+250x-2000$ .
b)
Find the prices that give a daily profit of $1000, and the number of mugs sold at each price.
c)
The formula gives $P(50)=-2000$ . What does the model predict at a price of $50, and why can't that happen?

Stuck? Peek at a hint

Profit is (profit per mug) $\times$ (number of mugs), and the profit per mug is $x-10$ dollars.

Think It Through (T)

Error Analysis · Building Equations

T1 Spot the Flaw

A student solves $3x^2+12x-5=0$ by completing the square.
Line 1: $3(x^2+4x)-5=0$
Line 2: $3(x^2+4x+4)-5-4=0$
Line 3: $3(x+2)^2=9$
Line 4: $(x+2)^2=3$
Line 5: $x+2=\sqrt{3}$ , so $x=-2+\sqrt{3}$ .

a)
Substitute the student's answer into $3x^2+12x-5$ . Is it a root?
b)
Two lines contain an error. Identify each one and correct it.
c)
Solve the equation correctly. Give the roots in exact form and to two decimal places.

Stuck? Peek at a hint

For (b), expand the student's Line 2 and compare it with $3x^2+12x-5$ .

T2 Build It Backwards

a)
Write a quadratic equation in standard form, with integer coefficients, whose roots are $\frac{3}{2}$ and $-4$ .
b)
Find every value of $k$ for which $x^2+kx+9=0$ has exactly one real root.
c)
Find the values of $k$ for which $2x^2-4x+k=0$ has no real roots.

Stuck? Peek at a hint

In (a), each root $r$ comes from a factor that equals $0$ at $r$ . In (b) and (c), write the discriminant in terms of $k$ .

Boss Level

Optional challenge

A Quadratic in Disguise

Consider the equation $(x^2-3x)^2-2(x^2-3x)-8=0$ .

a)
Let $u=x^2-3x$ . Rewrite the equation in terms of $u$ and solve for $u$ .
b)
Find all real solutions $x$ of the original equation.
c)
Find the value of $k$ for which $(x^2-3x)^2-2(x^2-3x)=k$ has exactly three real solutions.

Stuck? Peek at a hint

Treat $x^2-3x$ as a single block $u$ . In (c), count how many $x$ -values each value of $u$ produces.

Answer key

Final answers only. Try first, then check.

Core Skills

a) $x=4$ or $x=-\frac{2}{3}$
b) $x=0$ or $x=6$
c) $x=4$ or $x=-2$

a) $x=-2$ or $x=4$
b) $x=-1$ or $x=3$
c) $k=9$ : one solution ( $x=1$ ); $k=10$ : no solutions

a) $x=-5\pm\sqrt{7}$
b) No real roots: $(x-3)^2=-5$
c) $x=3\pm\dfrac{\sqrt{22}}{2}$

a) $x=2\pm\sqrt{10}$
b) $x=\dfrac{-1\pm\sqrt{7}}{3}$
c) $x=\dfrac{5\pm\sqrt{33}}{4}$ ; $x\approx2.69$ or $x\approx-0.19$

a) $D=0$ ; one real root
b) $D=-44$ ; no real roots
c) $D=41$ ; two real roots

Real World

a) $t\approx2.56$ s (the other root, $t\approx-0.16$ , is before the bump)
b) No: $5t^2-12t+8=0$ has $D=-16<0$ , so $h(t)=10$ has no solution.
c) $t\approx0.71$ s or $t\approx1.69$ s. The ball hits the ceiling at about $0.71$ s on the way up, so the model stops applying there: the landing at $2.56$ s (and the $1.69$ s root) never happens as predicted.

a) $(20-2x)(12-2x)=128$ , so $x^2-16x+28=0$
b) $x=2$ or $x=14$
c) $x=2$ : the path is $2$ m wide and the bed is $16$ m by $8$ m. $x=14$ is impossible: the bed's width would be $12-2(14)=-16$ m.

a) $n=200-5x$ ; $P(x)=(x-10)(200-5x)=-5x^2+250x-2000$
b) $20 (100 mugs) or $30 (50 mugs)
c) A loss of $2000, but that needs $n=200-5(50)=-50$ mugs sold, which is impossible. At $50 no mugs sell, so the model only applies for $x\le40$ .

Think It Through

a) The expression equals $-8$ , not $0$ , so $-2+\sqrt{3}$ is not a root.
b) Line 2: the $4$ inside the bracket is multiplied by $3$ , so it should read $-5-12=0$ . Line 5: a square root needs $\pm$ (and Line 3 becomes $3(x+2)^2=17$ ).
c) $x=-2\pm\dfrac{\sqrt{51}}{3}$ ; $x\approx0.38$ or $x\approx-4.38$

a) $2x^2+5x-12=0$ (or any non-zero integer multiple)
b) $k=6$ or $k=-6$
c) $k>2$

Boss Level

Boss

a) $u^2-2u-8=0$ , so $u=4$ or $u=-2$
b) $x=-1$ , $1$ , $2$ or $4$
c) $k=\dfrac{153}{16}$

Free cheat sheet

Quadratic Equations Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

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Most testedSolve $3x^2+5x-1=0$ . Give exact roots and roots to two decimal places. (card 4)

1Solving by factoring

$ab=0\Rightarrow a=0$ or $b=0$

UseMove every term to one side so it equals $0$ , factor, then set each factor equal to $0$ .

Example $3x^2-7x-6=0$ : $(3x+2)(x-3)=0$ , so $x=-2/3$ or $x=3$ .

Trap $(x-2)(x+4)=7$ is not $x-2=7$ : expand to $x^2+2x-15=0$ , so $x=3$ or $-5$ .

TrapDon't divide $x^2=5x$ by $x$ : you lose $x=0$ . Write $x(x-5)=0$ .

2Roots on a graph

UseThe roots of $ax^2+bx+c=0$ are the $x$ -intercepts of $y=ax^2+bx+c$ . To solve $ax^2+bx+c=k$ , read where the graph meets $y=k$ .

Example $x^2-2x-3=5$ : the graph meets $y=5$ at $x=-2$ and $x=4$ .

Trap $y=x^2-2x-3$ meets the $y$ -axis at $-3$ . That is the $y$ -intercept, not a root.

3Completing the square

UseRewrite as $a(x-h)^2+k=0$ , isolate the square, then take $\pm$ the square root. A negative on the other side: no real roots.

Example $x^2+6x+2=0$ : $(x+3)^2=7$ , so $x=-3\pm\sqrt7$ .

TrapKeep the $\pm$ : $(x-1)^2=9$ gives $x=4$ or $x=-2$ .

Trap $(x-2)^2=-3$ has no real roots; it does not give $x=2\pm\sqrt3$ .

4The quadratic formula

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

UseFirst write $ax^2+bx+c=0$ . Simplify the radical; round only at the end.

Example $x^2-2x-4=0$ : $x=(2\pm\sqrt{20})/2=1\pm\sqrt5$ , so $x\approx3.24$ or $-1.24$ .

Trap $2x^2-3x-4=0$ : $-b=+3$ and $b^2-4ac=9+32=41$ , not $9-32$ .

Trap $2a$ divides the whole top: $(2\pm\sqrt{20})/2=1\pm\sqrt5$ , not $2\pm\sqrt5$ .

5The discriminant

$D=b^2-4ac$ :

$D>0$	two real roots
$D=0$	one real root
$D<0$	no real roots

Trap $D=0$ means one root, not a root equal to $0$ : $x^2-6x+9=0$ has $x=3$ .

Trap $x^2+kx+16=0$ has one root when $k^2=64$ : $k=8$ or $k=-8$ .

6Word problems

UseSet up, rearrange to $=0$ , solve, then test each root: no negative lengths, times or counts.

ExampleA path inside a $10$ m by $6$ m garden leaves $32$ m $^2$ : $(10-2x)(6-2x)=32$ gives $x=1$ or $7$ ; only $x=1$ m fits.

TrapPositive is not enough: $x=7$ gives a width of $6-2(7)=-8$ .

Trap $D<0$ for a height of $10$ m: the ball never gets that high.

Which method?

It factors with small integers Factor

a=1

and

b

is even Complete the square

It doesn't factor Quadratic formula

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