Quadratic Equations: – Sum & Product of Roots – SS2 Further Mathematics Lesson Note (First Term)

For the quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.

• Sum of roots: α + β = -b/a.
• Product of roots: αβ = c/a.
• Equation from its roots: x² – (α + β)x + αβ = 0.
• Use: these results link an equation to its roots without solving it.

Behavioural objectives

By the end of the lesson, students should be able to:

1. derive the sum and product of the roots of ax² + bx + c = 0;
2. find the sum and product of the roots of a given quadratic equation;
3. form a quadratic equation when its roots are given;
4. find simple expressions such as α² + β² and 1/α + 1/β from the sum and product of roots.

What are the roots of a quadratic equation?

The roots of a quadratic equation are the values of x that make the equation true.

• Quadratic equation: an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
• Roots: the two values of x that satisfy the equation. We call them α (alpha) and β (beta).
• Discriminant (D): D = b² – 4ac. It tells us the nature of the roots.
• Quadratic formula: x = (-b ± √(b² – 4ac)) / 2a, which gives both roots.

What does the discriminant tell us?

How do we find the sum and product of the roots?

We find them by adding and multiplying the two roots given by the quadratic formula.

Let the roots of ax² + bx + c = 0 be:

• α: (-b + √D) / 2a
• β: (-b – √D) / 2a

Here D = b² – 4ac.

How do we derive the sum of the roots?

1. Write α + β = (-b + √D) / 2a + (-b – √D) / 2a.
2. Add the numerators over the common denominator: (-b + √D – b – √D) / 2a.
3. The √D terms cancel, leaving -2b / 2a.
4. Simplify: α + β = -b/a.

How do we derive the product of the roots?

1. Write αβ = (-b + √D)(-b – √D) / (2a × 2a).
2. The top is a difference of two squares: (-b)² – (√D)² = b² – D.
3. The bottom is 4a², so αβ = (b² – D) / 4a².
4. Put D = b² – 4ac: αβ = (b² – b² + 4ac) / 4a².
5. Simplify: αβ = 4ac / 4a² = c/a.

Hence, if ax² + bx + c = 0 and a ≠ 0, then α + β = -b/a and αβ = c/a.

How do we form a quadratic equation from its roots?

The quadratic equation whose roots are α and β is x² – (α + β)x + αβ = 0.

Why does this formula work?

1. Take x² + x – 42 = 0. It factorises as (x – 6)(x + 7) = 0, so the roots are 6 and -7.
2. In general, if (x – α)(x – β) = 0, then α and β are the roots.
3. Divide ax² + bx + c = 0 by a to get x² + (b/a)x + c/a = 0 … (i).
4. Expand (x – α)(x – β) = 0 to get x² – (α + β)x + αβ = 0 … (ii).
5. Compare the coefficients of (i) and (ii): -(α + β) = b/a, so α + β = -b/a.
6. Compare the constant terms: αβ = c/a.

What are the two types of problem?

• Type 1: given a quadratic equation, find the sum and product of its roots.
• Type 2: given the roots, form the quadratic equation.

What are the steps for forming an equation from its roots?

1. Find the sum of the roots, α + β.
2. Find the product of the roots, αβ.
3. Substitute into x² – (α + β)x + αβ = 0.
4. If there are fractions, multiply every term by the lowest common multiple of the denominators.

Worked examples

Example 1: Finding the sum and product of roots

Find the sum and product of the roots of each equation:

1. 2x² + 3x – 1 = 0
2. 3x² – 5x – 2 = 0
3. x² – 4x – 3 = 0
4. ½x² – 3x – 1 = 0

Solution

Read off a, b and c. Then use α + β = -b/a and αβ = c/a.

Working for (d):

• Sum: -b/a = 3 ÷ ½ = 6.
• Product: c/a = -1 ÷ ½ = -2.

Example 2: Forming the equation from its roots

Find the quadratic equation whose roots are:

1. 3 and -2
2. ½ and 5
3. -1 and 8
4. ¾ and ½

Solution

Use x² – (α + β)x + αβ = 0.

• Part (b): multiply every term by 2 to remove the fractions.
• Part (d): multiply every term by 8, the lowest common multiple of 4 and 8.

How do we find expressions in α and β?

We rewrite the expression using α + β and αβ, then substitute the known values.

• α² + β²: (α + β)² – 2αβ.
• 1/α + 1/β: (α + β) / αβ.
• (α – β)²: (α + β)² – 4αβ.

Example 3: Using the sum and product

If α and β are the roots of 2x² – 3x – 5 = 0, find (i) α² + β² and (ii) 1/α + 1/β.

Solution

1. a = 2, b = -3, c = -5.
2. α + β = -b/a = 3/2.
3. αβ = c/a = -5/2.
4. (i) α² + β² = (α + β)² – 2αβ = (3/2)² – 2(-5/2) = 9/4 + 5 = 29/4.
5. (ii) 1/α + 1/β = (α + β) / αβ = (3/2) ÷ (-5/2) = -3/5.

What mistakes should we avoid?

• Wrong sign for the sum: the sum is -b/a, so change the sign of b.
• Wrong sign in the formula: the equation is x² – (α + β)x + αβ = 0, so the middle sign is minus.
• Forgetting a: when a ≠ 1, divide b and c by a.
• Dividing by a fraction: to divide by ½, multiply by 2.
• Leaving fractions in the final equation: clear them by multiplying through.

Evaluation

1. State the sum and the product of the roots of ax² + bx + c = 0 in terms of a, b and c.
2. Write the quadratic equation whose roots are α and β.
3. Find the sum and product of the roots of 2x² + 5x – 3 = 0.
4. Find the quadratic equation whose roots are 4 and -1.

Assignment

Answer all parts:

1. Find the sum and product of the roots of 4x² – 8x + 3 = 0.
2. Find the sum and product of the roots of x² + 6x – 7 = 0.
3. Find the quadratic equation whose roots are 2 and 5.
4. Find the quadratic equation whose roots are -3 and ½.