Quadratic Equations · Learner pages

Lesson 1 · Build and recognise a quadratic equation

Convert a product situation into ax² + bx + c = 0.

Name or initials: ____________   Date: ____________

A square term x² comes from multiplying x by x.

1. Define the variable. Let a rectangle have width x metres and length x + 2 metres. Its area is 48 m².
2. Model the area. Area = length × width, so x(x + 2) = 48.
3. Write standard form. Expand and bring all terms to one side: x² + 2x − 48 = 0.
4. Keep the real situation. A solution used as a length must be positive. Algebraic roots must be checked against this condition.

Worked example: The rectangle equation factors as (x + 8)(x − 6) = 0. The roots are −8 and 6, but only width 6 m is positive. Length 8 m gives area 48 m².

1. A rectangle has width x and length x + 2. Its area is 3. Which equation models it?

   1. x² + 2x − 3 = 0 · 2. 2x + 2 − 3 = 0 · 3. x² − 2x − 3 = 0

2. A rectangle has width x and length x + 2. Its area is 8. Which equation models it?

   1. 2x + 2 − 8 = 0 · 2. x² − 2x − 8 = 0 · 3. x² + 2x − 8 = 0

3. A rectangle has width x and length x + 2. Its area is 15. Which equation models it?

   1. x² − 2x − 15 = 0 · 2. x² + 2x − 15 = 0 · 3. 2x + 2 − 15 = 0

Discuss: Model two consecutive positive integers whose product is 56, then identify the positive pair.

Lesson 2 · Solve by factorisation

Use the zero-product rule to retain both roots.

Name or initials: ____________   Date: ____________

A product can be zero even when only one factor is zero.

1. Put zero on one side. For x² = 9x − 20, rewrite as x² − 9x + 20 = 0.
2. Factor the expression. Two numbers with sum 9 and product 20 are 4 and 5. Thus (x − 4)(x − 5) = 0.
3. Separate two cases. Either x − 4 = 0 or x − 5 = 0. Keep both roots: 4 and 5.
4. Verify the roots. For x = 4, 16 = 36 − 20. For x = 5, 25 = 45 − 20. Both satisfy the original equation.

Worked example: Solve 2x² − 10x = 0. Factor 2x(x − 5) = 0. Thus x = 0 or x = 5. Checking gives 0 and 50 − 50 respectively.

1. Solve x² − 7x + 10 = 0.

   1. x = 2 or 5 · 2. x = -2 or -5 · 3. x = 0 or 7

2. Solve x² − 7x + 12 = 0.

   1. x = -3 or -4 · 2. x = 0 or 7 · 3. x = 3 or 4

3. Solve x² − 7x + 6 = 0.

   1. x = 0 or 7 · 2. x = 1 or 6 · 3. x = -1 or -6

Discuss: Make a quadratic whose roots are 0 and −4. Explain why dividing by x would lose the zero root.

Lesson 3 · Use the discriminant and formula

Decide the real-root type and calculate roots when D is nonnegative.

Name or initials: ____________   Date: ____________

The square of any real number is nonnegative.

1. Identify the signed coefficients. For 2x² − 7x + 3 = 0, a = 2, b = −7, c = 3.
2. Calculate D. D = (−7)² − 4 × 2 × 3 = 49 − 24 = 25. Positive D means two distinct real roots.
3. Use both signs. x = (7 ± 5)/4. The plus sign gives 3; the minus sign gives 1/2.
4. Interpret other signs. If D = 0 the roots coincide; if D < 0 there are no real roots. This does not mean the equation has no roots in every number system.

Worked example: For x² + 4x + 4 = 0, D = 16 − 16 = 0. The formula gives x = −4/2 = −2 twice, matching (x + 2)² = 0.

1. What is the nature of the roots of x² + (-6)x + (5) = 0?

   1. Two distinct real roots · 2. Two equal real roots · 3. No real roots

2. What is the nature of the roots of x² + (-4)x + (4) = 0?

   1. Two distinct real roots · 2. No real roots · 3. Two equal real roots

3. What is the nature of the roots of x² + (2)x + (5) = 0?

   1. Two equal real roots · 2. No real roots · 3. Two distinct real roots

Discuss: Compare x² − 4x + c = 0 for c = 3, 4 and 5. Use D to explain the three different real-root cases.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. A rectangle has width x and length x + 2. Its area is 35. Which equation models it?

   2x + 2 − 35 = 0 / x² − 2x − 35 = 0 / x² + 2x − 35 = 0

2. A rectangle has width x and length x + 2. Its area is 48. Which equation models it?

   x² − 2x − 48 = 0 / x² + 2x − 48 = 0 / 2x + 2 − 48 = 0

3. Solve x² − 10x + 16 = 0.

   x = -2 or -8 / x = 0 or 10 / x = 2 or 8

4. Solve x² − 14x + 45 = 0.

   x = 0 or 14 / x = 5 or 9 / x = -5 or -9

5. What is the nature of the roots of x² + (6)x + (9) = 0?

   Two distinct real roots / No real roots / Two equal real roots

6. What is the nature of the roots of x² + (4)x + (7) = 0?

   Two equal real roots / No real roots / Two distinct real roots

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. A rectangle has width x and length x + 2. Its area is 63. Which equation models it?

   x² + 2x − 63 = 0 / 2x + 2 − 63 = 0 / x² − 2x − 63 = 0

2. A rectangle has width x and length x + 2. Its area is 80. Which equation models it?

   2x + 2 − 80 = 0 / x² − 2x − 80 = 0 / x² + 2x − 80 = 0

3. Solve x² − 11x + 24 = 0.

   x = 3 or 8 / x = -3 or -8 / x = 0 or 11

4. Solve x² − 13x + 42 = 0.

   x = -6 or -7 / x = 0 or 13 / x = 6 or 7

5. What is the nature of the roots of x² + (-10)x + (21) = 0?

   Two distinct real roots / Two equal real roots / No real roots

6. What is the nature of the roots of x² + (2)x + (1) = 0?

   Two distinct real roots / No real roots / Two equal real roots