Solving Quadratic Equations — Drills
Solving Quadratic Equations has 8 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Area — Space inside, measured in square units.
- Perfect square — A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Factor — A number that divides evenly into another.
- Parabola — The U-shaped graph of a quadratic function.
- Roots — The x values where y = 0.
- Zero product property — If AB = 0 then A = 0 or B = 0.
- Quadratic formula — x = -b ± √b2 - 4ac2a.
- Discriminant — b2 - 4ac. Its sign counts the real solutions.
- Completing the square — Rewriting x2 + bx as a perfect square minus a constant.
Factoring
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve x2 + 4x - 12 = 0.
Worked answer
-6 and 2
(x + 6)(x - 2) = 0
x = -6 or x = 2
Problem 2
Solve x2 - 3x - 4 = 0.
Worked answer
-1 and 4
(x + 1)(x - 4) = 0
x = -1 or x = 4
Sqroot
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve (x - 1)2 = 16.
Worked answer
5 and -3
x - 1 = ± 4
x = 1 + 4 = 5 or x = 1 - 4 = -3
Problem 2
Solve x2 = 49.
Worked answer
7 and -7
x = ± 7
Factorout
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve 3x2 = 9x.
Worked answer
0 and 3
3x2 - 9x = 0
3x(x - 3) = 0
x = 0 or x = 3
Dividing by x would have lost the root 0.
Problem 2
Solve 2x2 = 12x.
Worked answer
0 and 6
2x2 - 12x = 0
2x(x - 6) = 0
x = 0 or x = 6
Dividing by x would have lost the root 0.
Formula
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve x2 + 3x - 4 = 0 using the quadratic formula.
Worked answer
-4 and 1
a = 1, b = 3, c = -4
b2 - 4ac = 9 - -16 = 25
x = -3 ± √252 = -3 ± 52
x = 1 or x = -4
Problem 2
Solve x2 - 3x - 4 = 0 using the quadratic formula.
Worked answer
-1 and 4
a = 1, b = -3, c = -4
b2 - 4ac = 9 - -16 = 25
x = 3 ± √252 = 3 ± 52
x = -1 or x = 4
Discrim
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the discriminant of 2x2 + 2x - 5 = 0 and say how many real solutions there are.
Worked answer
44, two real solutions
(2)2 = 4
4(2)(-5) = -40
4 - (-40) = 44
Positive, so two real solutions.
Problem 2
Find the discriminant of x2 - 1x + 5 = 0 and say how many real solutions there are.
Worked answer
-19, no real solutions
(-1)2 = 1
4(1)(5) = 20
1 - (20) = -19
Negative, so no real solutions.
Completing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve x2 + 10x = 39 by completing the square.
Worked answer
3 and -13
x2 + 10x + 25 = 64
(x + 5)2 = 64
x + 5 = ± 8
x = 3 or x = -13
Problem 2
Solve x2 + 14x = -13 by completing the square.
Worked answer
-1 and -13
x2 + 14x + 49 = 36
(x + 7)2 = 36
x + 7 = ± 6
x = -1 or x = -13
Zeroprod
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Given (x - 4)(x + 1) = 7, which step is valid?
- Set x - 4 = 7
- Expand and move everything to one side
- Set both factors to 7
- Set x + 1 = 7
Worked answer
Expand and move everything to one side
You cannot split a product that equals 7.
Expand first, then subtract to get zero on the right.
Problem 2
Given (x - 4)(x + 1) = 10, which step is valid?
- Set x + 1 = 10
- Expand and move everything to one side
- Set both factors to 10
- Set x - 4 = 10
Worked answer
Expand and move everything to one side
You cannot split a product that equals 10.
Expand first, then subtract to get zero on the right.
Word problems
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A rectangle is 7 cm longer than it is wide and has area 260 cm2. Find its width.
Worked answer
13
w(w + 7) = 260
w2 + 7w - 260 = 0
(w - 13)(w + 20) = 0
w = 13 cm — a width cannot be negative.
Problem 2
A rectangle is 6 cm longer than it is wide and has area 27 cm2. Find its width.
Worked answer
3
w(w + 6) = 27
w2 + 6w - 27 = 0
(w - 3)(w + 9) = 0
w = 3 cm — a width cannot be negative.