Nature of Roots and Formation of Quadratic Equations

In mathematics, a quadratic equation is a second-degree polynomial equation in a single variable, which can be written in the standard form:

ax² + bx + c = 0

where 'a', 'b', and 'c' are coefficients, and 'a' is not equal to zero. The 'roots' of a quadratic equation are the values of the variable 'x' that satisfy the equation. Understanding the nature of these roots is crucial for solving and interpreting quadratic equations.

The Discriminant

The nature of the roots of a quadratic equation is determined by a quantity called the discriminant, denoted by the Greek letter Delta (Δ). The discriminant is calculated using the coefficients of the quadratic equation as follows:

Δ = b² - 4ac

The value of the discriminant tells us about the roots without actually solving the equation. There are three main cases for the nature of the roots based on the discriminant:

Case 1: Δ > 0 (Discriminant is positive)

If the discriminant is positive (Δ > 0), the quadratic equation has two distinct real roots. This means there are two different real numbers for 'x' that will make the equation true.

Example: Consider the equation x² - 5x + 6 = 0. Here, a = 1, b = -5, and c = 6.

Calculate the discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1.

Since Δ = 1, which is greater than 0, the equation has two distinct real roots. We can verify this by solving: (x - 2)(x - 3) = 0, giving roots x = 2 and x = 3.

Case 2: Δ = 0 (Discriminant is zero)

If the discriminant is exactly zero (Δ = 0), the quadratic equation has exactly one real root, or two equal real roots. This is often referred to as a repeated root.

Example: Consider the equation x² - 4x + 4 = 0. Here, a = 1, b = -4, and c = 4.

Calculate the discriminant: Δ = (-4)² - 4(1)(4) = 16 - 16 = 0.

Since Δ = 0, the equation has one real root (or two equal real roots). Solving: (x - 2)(x - 2) = 0, giving the repeated root x = 2.

Case 3: Δ < 0 (Discriminant is negative)

If the discriminant is negative (Δ < 0), the quadratic equation has two complex roots. These roots are complex conjugates of each other. They are not real numbers.

Example: Consider the equation x² + x + 1 = 0. Here, a = 1, b = 1, and c = 1.

Calculate the discriminant: Δ = (1)² - 4(1)(1) = 1 - 4 = -3.

Since Δ = -3, which is less than 0, the equation has two complex roots. Using the quadratic formula, x = [-b ± sqrt(Δ)] / 2a, we get x = [-1 ± sqrt(-3)] / 2 = [-1 ± i*sqrt(3)] / 2. The roots are (-1 + i*sqrt(3))/2 and (-1 - i*sqrt(3))/2, which are complex conjugates.

Formation of Quadratic Equations

We can form a quadratic equation if we know its roots. Let the roots of a quadratic equation be α (alpha) and β (beta).

We know that the equation can be written in terms of its roots as:

(x - α)(x - β) = 0

Expanding this equation, we get:

x² - βx - αx + αβ = 0

x² - (α + β)x + αβ = 0

This gives us a standard form for constructing a quadratic equation from its roots:

x² - (Sum of Roots)x + (Product of Roots) = 0

This formula is fundamental. If we know the sum (α + β) and the product (αβ) of the roots, we can directly write down the quadratic equation.

Relationship between Roots and Coefficients

For a general quadratic equation ax² + bx + c = 0, if the roots are α and β, then we have the following relationships:

• Sum of Roots: α + β = -b/a
• Product of Roots: αβ = c/a

These relationships are derived by dividing the standard form of the quadratic equation by 'a':

x² + (b/a)x + (c/a) = 0

Comparing this with x² - (α + β)x + αβ = 0, we can see that:

-(α + β) = b/a => α + β = -b/a

αβ = c/a

Example of Forming a Quadratic Equation

Form a quadratic equation whose roots are 3 and 5.

Here, the roots are α = 3 and β = 5.

Sum of Roots: α + β = 3 + 5 = 8

Product of Roots: αβ = 3 * 5 = 15

Using the formula x² - (Sum of Roots)x + (Product of Roots) = 0:

x² - (8)x + 15 = 0

So, the quadratic equation is x² - 8x + 15 = 0.

Alternatively, we could use the form (x - α)(x - β) = 0:

(x - 3)(x - 5) = 0

x² - 5x - 3x + 15 = 0

x² - 8x + 15 = 0

Example with Complex Roots

Form a quadratic equation whose roots are 2 + 3i and 2 - 3i.

Here, α = 2 + 3i and β = 2 - 3i.

Sum of Roots: α + β = (2 + 3i) + (2 - 3i) = 4

Product of Roots: αβ = (2 + 3i)(2 - 3i)

This is in the form (a + bi)(a - bi) = a² + b².

αβ = 2² + 3² = 4 + 9 = 13

Using the formula x² - (Sum of Roots)x + (Product of Roots) = 0:

x² - (4)x + 13 = 0

The quadratic equation is x² - 4x + 13 = 0.

Example using Coefficients

Form a quadratic equation with real coefficients whose roots are 1 + sqrt(2) and 1 - sqrt(2).

Here, α = 1 + sqrt(2) and β = 1 - sqrt(2).

Sum of Roots: α + β = (1 + sqrt(2)) + (1 - sqrt(2)) = 2

Product of Roots: αβ = (1 + sqrt(2))(1 - sqrt(2))

This is in the form (a + b)(a - b) = a² - b².

αβ = 1² - (sqrt(2))² = 1 - 2 = -1

Using the formula x² - (Sum of Roots)x + (Product of Roots) = 0:

x² - (2)x + (-1) = 0

The quadratic equation is x² - 2x - 1 = 0.

Notice that if a quadratic equation has real coefficients, then any complex roots must occur in conjugate pairs, and any irrational roots involving square roots must also occur in conjugate pairs (of the form a + sqrt(b) and a - sqrt(b)).

Quadratic Equations with Non-Real Coefficients

The rules for forming quadratic equations from roots still apply even if the coefficients are not real. However, if the coefficients are not real, the roots do not necessarily have to be complex conjugates.

Example: Form a quadratic equation whose roots are 2i and 3.

Here, α = 2i and β = 3.

Sum of Roots: α + β = 2i + 3 = 3 + 2i

Product of Roots: αβ = (2i)(3) = 6i

Using the formula x² - (Sum of Roots)x + (Product of Roots) = 0:

x² - (3 + 2i)x + 6i = 0

This quadratic equation has complex coefficients.

Special Cases and Considerations

Equations Reducible to Quadratic Form

Sometimes, equations that are not initially quadratic can be transformed into quadratic equations. This often involves a substitution.

Example: Solve x⁴ - 5x² + 4 = 0.

Let y = x². Then the equation becomes:

y² - 5y + 4 = 0

This is a quadratic equation in 'y'. We can solve it by factoring:

(y - 1)(y - 4) = 0

So, y = 1 or y = 4.

Now, substitute back x² for y:

If y = 1, then x² = 1, which gives x = ±1.

If y = 4, then x² = 4, which gives x = ±2.

The roots of the original equation are 1, -1, 2, and -2.

Common Factors and Roots

If two quadratic equations share a common root, say 'k', then substituting 'k' into both equations will result in 0.

Let the two equations be: a₁x² + b₁x + c₁ = 0 a₂x² + b₂x + c₂ = 0

If they share a common root 'k', then: a₁k² + b₁k + c₁ = 0 a₂k² + b₂k + c₂ = 0

To find the condition for a common root, we can use various algebraic methods, such as elimination or determinants.

One method is to eliminate the k² term. Multiply the first equation by a₂ and the second by a₁: a₁a₂k² + b₁a₂k + c₁a₂ = 0 a₁a₂k² + b₂a₁k + c₂a₁ = 0

Subtracting the second from the first: (b₁a₂ - b₂a₁)k + (c₁a₂ - c₂a₁) = 0

If (b₁a₂ - b₂a₁) is not zero, then: k = - (c₁a₂ - c₂a₁) / (b₁a₂ - b₂a₁) k = (c₂a₁ - c₁a₂) / (a₁b₂ - a₂b₁)

This gives the value of the common root. For this root 'k' to exist, it must satisfy the original equations. Substituting this value of 'k' back into one of the equations can lead to the condition for the common root.

A more elegant way to find the condition for a common root is using determinants. The condition for the quadratic equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 to have a common root is:

(c₁a₂ - c₂a₁)² = (a₁b₂ - a₂b₁)(b₁c₂ - b₂c₁)

This condition ensures that there is a value of 'x' that satisfies both equations simultaneously.

Coincident Equations

Two quadratic equations are coincident if they have the same roots. This means their coefficients are proportional.

a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 are coincident if:

a₁/a₂ = b₁/b₂ = c₁/c₂

This is a special case of having common roots, where both roots are common.

Summary of Key Concepts

Understanding the nature of roots and how to form quadratic equations is fundamental.

• The discriminant (Δ = b² - 4ac) determines the nature of roots:
  • Δ > 0: Two distinct real roots.
  • Δ = 0: One real root (repeated).
  • Δ < 0: Two complex conjugate roots.
• A quadratic equation can be formed from its roots (α, β) using the formula: x² - (α + β)x + αβ = 0.
• For ax² + bx + c = 0, the sum of roots is -b/a and the product of roots is c/a.
• Equations reducible to quadratic form can be solved using substitution.
• The condition for two quadratic equations to have a common root involves a specific algebraic relationship between their coefficients.