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Lesson 4 of 4

Quadratic Equations · Lesson 4 of 4

Chapter Summary and Practice

“Roots, factorisation and the quadratic formula reunite for one final equation-solving workout.”

Learning Objectives

• Recall the standard form and meaning of a quadratic equation. • Choose between factorisation and the quadratic formula. • Use the discriminant to determine the nature of roots. • Solve mixed quadratic-equation problems step by step. • Interpret roots correctly in real situations.

Quadratic equations connect algebra with many real situations. The essential skills are recognising a quadratic equation, forming one from a situation, solving it by a suitable method, and interpreting the roots correctly.

Quadratic Equation

Standard formLaTeX

Always simplify first. An equation that appears quadratic may become linear after cancellation, while an equation that appears cubic may simplify to degree 2.

Roots

A root is a value of x that makes ax² + bx + c equal to zero. A quadratic equation can have at most two real roots.

Factorisation

If ax² + bx + c can be written as two linear factors, solve by setting each factor equal to zero.

Zero-product propertyLaTeX

Quadratic Formula

Quadratic formulaLaTeX

Nature of Roots

DiscriminantLaTeX
ConditionNature of roots
D > 0Two distinct real roots
D = 0Two equal real roots
D < 0No real roots

How to Choose the Right Method

SituationBest first approach
Expression factorises easilyFactorisation
No obvious factor pairQuadratic formula
Only nature of roots is requiredCalculate the discriminant
A parameter must give equal rootsSet D = 0
Real-world situationForm the equation, solve, then reject impossible roots
Common Mistakes

• Deciding whether an equation is quadratic before simplifying it, • Forgetting a ≠ 0, • Forcing factorisation when no simple factor pair exists, • Losing the sign of b in the quadratic formula, • Using the wrong discriminant formula, • Keeping a negative root when it represents an impossible length, age or count,

Guided Practice

Guided Example 1: Recognise the Equation

Problem
Determine whether (x + 3)² = x² + 4x + 10 is quadratic.

  1. 1.Expand the left side: x² + 6x + 9.
  2. 2.Set equal to x² + 4x + 10.
  3. 3.Cancel x² from both sides.
  4. 4.2x − 1 = 0.
  5. 5.The simplified equation is linear, not quadratic.
Guided Example 2: Factorisation

Problem
Solve x² − 3x − 10 = 0.

  1. 1.Find two numbers with product −10 and sum −3: −5 and 2.
  2. 2.x² − 5x + 2x − 10 = 0.
  3. 3.x(x − 5) + 2(x − 5) = 0.
  4. 4.(x + 2)(x − 5) = 0.
  5. 5.So x = −2 or x = 5.
Guided Example 3: Quadratic Formula

Problem
Solve x² − 5x + 5 = 0.

  1. 1.a = 1, b = −5, c = 5.
  2. 2.D = 25 − 20 = 5.
  3. 3.x = [5 ± √5]/2.
  4. 4.So the roots are (5 + √5)/2 and (5 − √5)/2.
Guided Example 4: Nature of Roots

Problem
Find the nature of roots of 2x² − 4x + 3 = 0.

  1. 1.D = (−4)² − 4(2)(3).
  2. 2.D = 16 − 24 = −8.
  3. 3.Since D < 0, there are no real roots.
Guided Example 5: Real Situation

Problem
Two consecutive positive integers have product 306. Find them.

  1. 1.Let the smaller integer be x, so the next is x + 1.
  2. 2.x(x + 1) = 306.
  3. 3.x² + x − 306 = 0.
  4. 4.Factorise: (x − 17)(x + 18) = 0.
  5. 5.x = 17 or x = −18.
  6. 6.The integers are positive, so x = 17.
  7. 7.Therefore the numbers are 17 and 18.

Practise Problems

Practice Problems

Practice Questions
  1. Check whether (x + 1)² = 2(x − 3) is quadratic after simplification.
  2. Check whether (x − 3)(2x + 1) = x(x + 5) is quadratic.
  3. A rectangular plot has area 528 m² and length 1 m more than twice the breadth. Form the quadratic equation.
  4. Solve 2x² + x − 6 = 0 by factorisation.
  5. Solve 100x² − 20x + 1 = 0 by factorisation.
  6. Find two numbers whose sum is 27 and product is 182.
  7. Find two consecutive positive integers whose squares add to 365.
  8. Find the nature of roots of 2x² − 3x + 5 = 0.
  9. Find the nature and roots of 2x² − 6x + 3 = 0.
  10. Find k so that 2x² + kx + 3 = 0 has equal roots.
  11. Determine whether a rectangle of perimeter 80 m and area 400 m² is possible.
  12. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages was 48. Form and solve the quadratic equation.

Quick Mixed Check

Quiz

Quick check

Which is the standard form of a quadratic equation?

Quick check

For ax² + bx + c = 0, the discriminant is:

Quick check

If D = 0, the equation has:

Quick check

If (x − 4)(x + 2) = 0, the roots are:

Quick check

If D < 0, what can we conclude over the real numbers?

Before You Finish the Chapter

Make sure you can simplify before classifying, form equations from words, split the middle term, use the zero-product property, apply the quadratic formula carefully, and interpret the discriminant before solving.

Key Takeaways

Key Takeaways

• Quadratic equations have standard form ax² + bx + c = 0 with a ≠ 0. • Roots make the quadratic expression equal to zero. • Factorisation is efficient when the expression splits neatly. • The quadratic formula works more generally. • The discriminant reveals the nature of roots before full calculation.

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Nature of Roots

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