Solving Monic Quadratic Equations
Solutions to Quadratics
Draw a line from each equation to its pair of solutions.
Check the Solution
Circle the correct solution(s) for each equation.
x^2 - 10x + 21 = 0
x^2 + 2x - 15 = 0
x^2 - 16 = 0
Solve by Factorising
Solve each quadratic equation by factorising. Show all steps.
x^2 + 9x + 20 = 0
x^2 - 8x + 15 = 0
x^2 + 3x - 28 = 0
x^2 - 11x + 30 = 0
Rearrange then Solve
Rearrange each equation into the form x^2 + bx + c = 0 before factorising.
x^2 + 4x = 12
x^2 = 7x - 10
x^2 - 5x = -6
Verify Your Solutions
Substitute each solution back into the original equation to verify.
For x^2 - 7x + 12 = 0, you found x = 3 and x = 4. Verify both solutions:
Problem Solving with Quadratics
Set up and solve a quadratic equation for each problem.
The product of two consecutive integers is 56. Let the smaller integer be n. Write and solve the quadratic equation to find both integers.
A square has its length increased by 3 cm and its width decreased by 1 cm, giving an area of 40 cm^2. Let the original side length be x cm. Write and solve an equation to find x.
Graphical Connection
Connect your algebra to the graph.
If x^2 - 3x - 10 = 0 has solutions x = 5 and x = -2, what are the x-intercepts of the parabola y = x^2 - 3x - 10? Explain your reasoning.
Quadratic Equations at Home
Investigate quadratics in everyday contexts.
- 1Look up how quadratic equations describe projectile motion (e.g. a ball thrown in the air). Write one example equation and describe what each variable represents.
- 2Create a number puzzle: I am thinking of two numbers. They multiply to 24 and add to 11. Give it to a family member and show how factorising solves it.
Null Factor Law Practice
Use the null factor law: if A x B = 0, then A = 0 or B = 0. Find all solutions.
(x + 1)(x - 6) = 0 -> x = ___ or x = ___
(2x - 4)(x + 3) = 0 -> x = ___ or x = ___
x(x - 9) = 0 -> x = ___ or x = ___
(x - 5)^2 = 0 -> x = ___ (explain why there is only one solution)
Checking Solutions by Substitution
Substitute each claimed solution back into the equation to verify it is correct.
Equation: x^2 - 6x + 8 = 0. Check x = 2: LHS = ___. Check x = 4: LHS = ___.
Equation: x^2 + x - 12 = 0. Check x = 3: LHS = ___. Check x = -4: LHS = ___.
Forming Quadratics from Given Roots
If x = p and x = q are the roots of a quadratic, then the equation is (x - p)(x - q) = 0. Write the quadratic equation for each pair of roots.
Roots: x = 3 and x = 5. Equation:
Roots: x = -2 and x = 7. Equation:
Roots: x = -4 and x = -1. Equation:
Root: x = 6 (double root). Equation:
Word Problems Leading to Quadratics
Write a quadratic equation for each problem, solve it, and check your answer makes sense.
A rectangle has a length 3 cm greater than its width. Its area is 40 cm^2. Let the width = x. Find the dimensions.
The sum of a number and its square is 42. Find the number.
How Many Solutions?
Decide how many real solutions each equation has, without solving fully.
x^2 - 9 = 0
x^2 + 4 = 0
(x - 3)^2 = 0
x^2 - 5x + 6 = 0
Solve by Rearranging First
These equations are not in standard form. Rearrange, then factorise and solve.
x^2 = 5x + 24. Solution:
x^2 + 3x = 40. Solution:
x(x - 7) = -12. Solution:
(x + 3)(x - 4) = 8. Solution:
Quadratic Applications — Motion and Area
Set up and solve quadratic equations for each scenario.
A ball is thrown upward. Its height is h = -t^2 + 6t metres. When is it 5 metres high? (Set h = 5 and solve.)
A rectangle has width x and length (x+5). Its area is 84 m^2. Find its dimensions.
Two positive integers differ by 3. Their product is 54. Find the integers.
Solve Using the Quadratic Formula — Preview
Use x = (-b ± sqrt(b^2 - 4c)) / 2 to solve these equations that cannot be factorised neatly.
x^2 - 4x + 1 = 0. Discriminant = b^2 - 4c = ___. Solutions x = (4 ± sqrt(___)) / 2 = ___ ± ___
x^2 + 2x - 5 = 0. Use the formula to find exact solutions.
Why do these equations have irrational solutions, whereas x^2 - 5x + 6 = 0 has rational solutions?
Error Analysis — Quadratic Equations
Each solution below contains an error. Find and correct it.
Student solves x^2 + 5x = 6 by writing x(x+5) = 6, so x = 1 or x+5 = 6 → x = 1. Error:
Student writes: x^2 - 36 = 0, so (x-6)^2 = 0, so x = 6 only. Error:
Student writes: x^2 + 4x + 4 = 0 → (x+4)^2 = 0 → x = -4. Error:
Classify Quadratic Equations
Sort each equation by the number of real solutions.
Quadratic Equations — Worded Challenges
Model and solve each problem.
A number is 4 more than its reciprocal times 6. Write and solve the quadratic.
A square garden has a path of width 1 m all around it. The total area (garden + path) is 100 m^2. Find the side of the garden.
The product of two consecutive negative integers is 90. Find both integers.
Quadratics in the Real World
Find and explore quadratic equations in everyday contexts.
- 1Research the trajectory formula for a basketball: h = h0 + v*t - 4.9t^2. Substitute real values and find when the ball reaches a given height.
- 2Look up 'break-even analysis in business'. Revenue and cost curves produce quadratic profit functions. Find an example online and solve the break-even equation.
- 3Use desmos.com to graph y = x^2 - 5x + 6. Observe where it crosses the x-axis. Change the equation and predict the new x-intercepts, then verify.
Quadratic Equations — Extension: Vieta's Formulas
For x^2 + bx + c = 0 with roots p and q, it can be shown that p + q = -b and pq = c.
For x^2 - 7x + 10 = 0: predict p + q = ___ and pq = ___ without solving. Then verify by solving.
A quadratic has roots 3 and -5. Without expanding fully, state its coefficients b and c using Vieta's Formulas.
If the sum of the roots of x^2 + bx + 12 = 0 is -7, find b and the two roots.
Quadratic Equations — Comprehensive Exam Practice
Work independently for 25 minutes. Show all steps.
Solve: x^2 - 13x + 36 = 0
Solve: x^2 + 5x = 24 (rearrange first)
Solve: (x - 2)^2 = 25
Form and solve a quadratic: three times the square of a number minus the number equals 14.
A square lawn has perimeter increased by 4 m on each side, making a larger square of area 169 m^2. Find the original side length.
Explain in 2-3 sentences how factorisation, the zero product property and roots are all connected.
Solve Using Factorisation — Set A
Solve each equation by factorising. Show all working.
x^2 - 10x + 24 = 0
x^2 + 9x + 20 = 0
x^2 - 4x - 12 = 0
x^2 - 49 = 0
x^2 - 12x + 36 = 0
Rearrange and Solve
Rearrange each equation to standard form, then solve by factorising.
x^2 = 9x - 14. Steps: ___, ___, ___. Solution:
x^2 + 2x = 48. Steps: ___, ___, ___. Solution:
x(x + 3) = 28. Steps: ___, ___, ___. Solution:
(x - 4)(x + 2) = 16. Steps: ___, ___, ___. Solution:
Solve Quadratics — Developing Difficulty
These increase in difficulty. Show all steps.
x^2 - x - 56 = 0
x^2 + 2x - 99 = 0
2x^2 - 8x - 10 = 0. Hint: divide through by 2 first.
3x^2 + 15x + 18 = 0. Hint: factor out the common factor first.
x^2 - 20x + 100 = 0
Correct Solution Method
Circle the method that correctly solves each equation.
x^2 - 7x = 0
x^2 = 36
x^2 + 8x + 16 = 0
Quadratics in Geometry — Area Problems
Write and solve a quadratic equation for each area problem.
A rectangle has area 45 cm^2. Its length is x + 4 and width is x. Find the dimensions.
A right triangle has legs x and (x + 3). Its area is 14 cm^2. Find x.
A square is enlarged by adding 5 m to each side. The new area is 169 m^2. Find the original side length.
Two squares have side lengths x and (x + 4). The larger area minus the smaller area is 80. Find x.
Forming Quadratics from Number Conditions
Write a quadratic equation for each number relationship and solve.
A number exceeds its square by 12 times itself. Find the number.
The square of a number is 17 more than 16 times the number. Find all solutions.
The product of (n + 3) and (n - 3) is 7. Find n.
The square of the sum of two consecutive integers is 121. Find the integers.
Quadratics in Speed and Distance
Solve these motion problems using quadratic equations.
A train travels x km at a speed of (x + 10) km/h. The trip takes 2 hours. Write and solve the quadratic (distance = speed x time). Find x.
A car travels 240 km. If the speed were 20 km/h faster, the trip would take 1 hour less. If speed = x, write and solve the quadratic equation.
Quadratic Equations — Consolidating Set
Solve each equation and check your solutions.
x^2 - 8x = 0
x^2 - 18x + 81 = 0
x^2 - 5x - 84 = 0
x(x + 12) = 45
(x - 1)^2 = 49
2x^2 + 14x + 24 = 0
Sort Solutions by Number of Real Roots
Place each equation in the correct column based on the number of real solutions.
Quadratic Equations at Home
Apply quadratic equation solving to real-life situations.
- 1Measure a rectangular space (room, garden bed, bookshelf). Assign x to one dimension and express the other in terms of x. Calculate the area and set it equal to the actual area to verify your equation works.
- 2Research 'golden ratio and quadratic equations'. The golden ratio phi satisfies phi^2 = phi + 1. Rearrange this to x^2 - x - 1 = 0 and solve it using the quadratic formula. You should get phi = (1 + sqrt(5))/2.
- 3Use desmos.com to graph y = x^2 - 5x + 4. Read the x-intercepts from the graph, then verify algebraically by solving x^2 - 5x + 4 = 0 by factorising.
Connecting Quadratic Equations to Quadratic Functions
Explore the relationship between solving a quadratic equation and graphing a quadratic function.
Solve x^2 - 4x + 3 = 0 by factorising. Write the solutions.
If y = x^2 - 4x + 3, at what x-values does the parabola cross the x-axis?
Solve x^2 - 4x + 4 = 0. How many x-intercepts does y = x^2 - 4x + 4 have?
Solve x^2 + 4 = 0. Can the parabola y = x^2 + 4 ever reach the x-axis? Explain.
Derive the Quadratic Formula
Work through the derivation of the quadratic formula by completing the square.
Start with ax^2 + bx + c = 0. Divide through by a:
Move c/a to the right side:
Add (b/2a)^2 to both sides to complete the square:
Write the left side as a perfect square:
Take the square root of both sides and solve for x:
You should have x = (-b ± sqrt(b^2 - 4ac)) / 2a. Does your derivation match?
Quadratic Equations — Comprehensive Final Exam
Work for 30 minutes independently. Show all working and check solutions.
Solve: x^2 + 3x - 40 = 0
Solve: 5x^2 - 20 = 0
Solve: (x + 3)^2 = 49
A rectangle has length 2 more than its width. Its area is 80 cm^2. Find the dimensions.
Two positive numbers differ by 5. Their product is 126. Find both numbers.
Solve x^2 - 2x - 35 = 0 graphically by sketching y = x^2 - 2x - 35 and finding x-intercepts, then confirm algebraically.
Explain the connection between factorising a quadratic and solving the equation. Why does 'setting each factor to zero' work?
Graphical and Algebraic Solutions — Side by Side
For each quadratic, find the roots algebraically and verify graphically.
y = x^2 - 4x + 3. Solve algebraically: roots = ___. Sketch and label.
y = x^2 - 1. Solve algebraically: roots = ___. Sketch and label.
y = x^2 + 4x. Solve algebraically: roots = ___. Sketch and label.
Solve Quadratics — Extending Set
These require more steps. Show all working carefully.
4x^2 - 36x + 56 = 0. Factor out 4 first.
6x^2 + 18x - 60 = 0. Factor out 6 first.
x^2 - 11x = 0
(x + 5)^2 - 9 = 0. Expand and rearrange, then solve.
x^2/4 - x + 1 = 0. Multiply through by 4 first.
Quadratic Equations — Applied Scenarios
Write and solve a quadratic for each scenario.
A garden bed is x metres wide and (x + 7) metres long. A fence costs $20 per metre around the perimeter. If the fencing budget is $200, find x.
A ball is thrown from a roof 20 m high with upward velocity 10 m/s. Its height is h = -5t^2 + 10t + 20. When does it reach the ground?
The ages of two siblings differ by 4 years. The product of their ages is 96. Find their current ages.
Match Equation to Solution
Match each quadratic equation to its solution set.
Self-Created Word Problem
Create your own word problem that leads to a quadratic equation, solve it, and verify.
Write your word problem (it must involve a quadratic):
Write and solve the quadratic equation:
Verify your solution in the context of the problem:
Investigate: Sum and Product of Roots
Explore Vieta's formulas through examples.
Solve x^2 - 5x + 6 = 0. Sum of roots = ___. Product of roots = ___. Check: b = -5, c = 6. Sum = ___? Product = ___?
Solve x^2 + 7x - 18 = 0. Sum of roots = ___. Product of roots = ___. Check against b and c.
Generalise: for x^2 + bx + c = 0, sum of roots = ___ and product = ___. Explain in words why this must be true using (x-p)(x-q) expansion.
Quadratic Equations in Technology
Investigate quadratic equations in science and engineering applications.
- 1Look up 'ballistic trajectory calculator' online. Enter initial velocity = 20 m/s and angle = 30 degrees. Record the time of flight. Verify: for h = 0, the quadratic formula gives this time.
- 2Use Desmos to graph y = -x^2 + 10x. Read the vertex (highest point) and x-intercepts. Verify by solving -x^2 + 10x = 0.
- 3Research stopping distance for vehicles: d = v + v^2/20 (v in km/h). Solve for v when d = 30 m. You will need to solve a quadratic.
Quadratic Equations — Unit Reflection
Reflect on everything you have learned about solving quadratic equations.
List the methods for solving quadratic equations that you know. When would you use each?
What was the hardest type of problem in this worksheet? How did you work through it?
Solve three more quadratics from memory, each using a different method.
Rate your confidence (1-10) in solving quadratic equations for an exam.
Quadratics — Writing for a Younger Student
Imagine explaining quadratic equations to a Year 8 student. Write clear instructions.
Explain: what is a quadratic equation? How is it different from a linear equation?
Explain: what does 'solving' a quadratic equation mean?
Write step-by-step instructions for solving x^2 - 7x + 12 = 0 that a Year 8 could follow.
Explain why there can be two solutions, one solution, or no real solutions.
Solve Quadratics — Speed and Accuracy Round
Solve these 10 quadratics. Aim for under 15 minutes.
1. x^2 - 8x + 15 = 0 2. x^2 + 3x - 18 = 0
3. x^2 - 25 = 0 4. x^2 + 6x + 9 = 0
5. x^2 - 15x + 56 = 0 6. x^2 + 2x - 48 = 0
7. x^2 - 64 = 0 8. x^2 + 14x + 45 = 0
9. 2x^2 - 18 = 0 10. x^2 - 19x + 88 = 0
Quadratic Equations — Advanced Problem Set
These problems require careful algebraic setup before solving.
A farmer has 80 m of fencing to enclose a rectangular paddock against a straight wall (no fence needed on the wall side). The paddock's area must be 750 m^2. Let the width = x. Set up and solve the quadratic.
A train travels 300 km. If it went 10 km/h faster it would save 1 hour. If speed = x, form and solve the quadratic equation.
Mia is 3 years older than Tom. In 5 years, the product of their ages will be 130. Find their current ages.
Correct Setup for Word Problems
Circle the correct quadratic equation for each scenario.
Width x, length x+3, area 40
Two consecutive integers, product 72
Number squared equals 5 more than 4 times itself
Quadratic Equations — Absolute Final Review
Complete every question. This is the final preparation for the next topic.
State the zero product property and explain why it is needed to solve quadratics by factorisation.
Solve: x^2 - 10x + 21 = 0
Solve: x^2 = 4x + 12
A rectangle has area 42 cm^2. Its width is (x - 1) and length is (x + 6). Find x and the dimensions.
Use the discriminant to determine whether x^2 - 6x + 10 = 0 has real solutions. Explain your answer.
Create a quadratic equation with roots 4 and -9. Expand it to standard form.
Quadratics in Science: Drug Concentration
A medication enters the bloodstream. Its concentration c(t) = -t^2 + 6t mg/L at time t hours.
When is the concentration zero? Set c = 0 and solve the quadratic.
What is the maximum concentration and when does it occur? (Use t = average of roots.)
For what time interval is the concentration above 5 mg/L? Set c = 5 and solve.
Identify the Correct Solution
Circle the fully correct solution.
x^2 = 16
x(x+6) = 0
(x-5)(x+2) = 0
x^2 - 3x = 0
Quadratic Equations — Extended Thinking
Answer these conceptual questions about quadratic equations.
Why does the zero product property require the product to equal zero specifically? What goes wrong if you apply it to (x-3)(x+2) = 6?
A quadratic always has at most two solutions. Why can't it have three? (Think about the degree of the polynomial.)
Describe three different real-world situations that produce quadratic equations. For each, explain what the two solutions represent.
Final 15 Quadratic Equations
Solve each equation. Mark your time and work accurately.
1. x^2 + 11x + 28 = 0 2. x^2 - 11x + 28 = 0 3. x^2 + 3x - 28 = 0
4. x^2 - 3x - 28 = 0 5. x^2 - 196 = 0 6. x^2 - 16x + 64 = 0
7. x^2 + 16x + 64 = 0 8. x(x - 11) = 0 9. 3x^2 - 27 = 0
10. x^2 = 11x - 30 11. (x+4)^2 = 81 12. x^2 + 18x + 81 = 0
13. 2x^2 - 50 = 0 14. x^2 - x - 72 = 0 15. x^2 + x - 72 = 0
Quadratic Equations — Proof and Generalisation
Use algebra to prove general results about quadratic equations.
For x^2 + bx + c = 0 with roots p and q, expand (x-p)(x-q) to show that p + q = -b and pq = c.
If a quadratic has two roots that differ by 4, what can you say about its discriminant? Write an equation relating b, c, and the condition 'roots differ by 4'.
Quadratic Equations — Capstone Challenge
This final problem combines multiple skills. Read carefully before starting.
Sasha walks x km north then (x+3) km east, ending at a point 15 km from start. (a) Write an equation using Pythagoras' theorem. (b) Expand and simplify to standard quadratic form. (c) Solve by factorising. (d) State the distances she walked north and east.
A quadratic equation has roots that are consecutive even integers. The sum of the roots is 14. (a) Find the roots. (b) Write the quadratic equation in standard form.
Quadratic Summary and Knowledge Map
Create a knowledge map of quadratic equations.
In the centre write 'Quadratic Equation'. Branch out to: Standard Form, Factorisation, Zero Product Property, Solutions/Roots, Discriminant, Parabola. For each branch, add a key fact or example.
Connect at least three pairs of branches with a relationship arrow and label the connection.
Quadratics Mastery — Self-Assessment
Honest self-assessment before moving to the next topic.
Which part of solving quadratic equations is completely automatic for you now?
Which step do you still sometimes make errors on? Write the error you made and the correct approach.
Solve three quadratics from memory right now to demonstrate your mastery:
What connections do you see between this topic and factorisation, graphing, and real-world problems?
Quadratic Equations — Final Five
Solve each equation completely. Show all four steps.
x^2 - 12x + 35 = 0
x^2 + 4x = 21
x^2 - 196 = 0
x^2 + 4x + 4 = 0
5x^2 - 45 = 0
Which Statement is Correct?
Circle the correct statement.
For x^2 - bx + c = 0 with positive roots p, q:
The discriminant of x^2 + 5x + 4 = 0 is:
x^2 - 9x + 20 = 0 has solutions:
Match Equation to Root Pair
Match each quadratic equation to its pair of roots.
Quadratic Mastery — Final Five Questions
Complete these five challenging questions to confirm mastery.
Solve: x^2 - 2x - 99 = 0
Solve: 4x^2 - 100 = 0 (factor out the common factor first)
Form and solve: the product of two numbers is 56. One number is 1 more than the other. Find both.
Show algebraically that x^2 - 12x + 36 has exactly one root and it is x = 6.
Use Vieta's formulas to find the sum and product of the roots of x^2 - 14x + 45 = 0 without solving. Then solve to verify.