Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of degree 2 in one variable. For example, x² - 11x + 28, 3x² - 11x + 23 = 0, etc.

Its standard form is:

• Where a, b, and c are constants, x is the variable, and a ≠ 0.

The values of x that satisfy the equation are called the roots (or solutions) of the quadratic equation

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of x that make the equation equal to zero. In simple terms, roots are the solutions of the quadratic equation.

A quadratic equation is written in the standard form ax² + bx + c = 0, where a ≠ 0.

The values of x that satisfy this equation are called the roots or zeros of the quadratic equation. These roots also represent the points where the graph of the quadratic function (a parabola) intersects the x-axis.

• Since a quadratic equation is a second-degree polynomial, it can have at most two roots. The roots may be real or complex, and in some cases, both roots can be equal.

Example: Consider the quadratic equation x² − 3x − 4 = 0

The roots of this equation are x = −1 and x = 4, because both values satisfy the equation.

For x = −1
(−1)² − 3(−1) − 4 = 1 + 3 − 4 = 0

For x = 4
(4)² − 3(4) − 4 = 16 − 12 − 4 = 0

Quadratic Formula

The quadratic formula is a method used to find the roots of a quadratic equation, especially when the equation cannot be easily factored. For a quadratic equation of the form ax² + bx + c = 0, the roots are given by:

The ± sign gives the two possible roots of the equation. This formula is also known as the Sridharacharya formula.

Example: Find the roots of the quadratic equation x² − 3x − 4 = 0 using the quadratic formula.

Solution:

Here, a = 1 ,b = −3 and c = −4

Using the quadratic formula:
x = (-b ± √(b² − 4ac)) / 2a

Substitute the values:

x = [−(−3) ± √((−3)² − 4(1)(−4))] / 2(1)

= [3 ± √(9 + 16)] / 2

= [3 ± √25] / 2

= (3 + 5) / 2 or (3 − 5) / 2

= 8/2 or −2/2

x = 4 or x = −1

Therefore, the roots of the equation are 4 and −1.

Nature of Roots

The nature of the roots of a quadratic equation can be determined without actually solving the equation. This is done using the discriminant.

For a quadratic equation ax² + bx + c = 0, the discriminant is given by:

The value of the discriminant helps us understand the type of roots of the equation.

• If D > 0 → The equation has two real and distinct roots.
• If D = 0 → The equation has two real and equal roots.
• If D < 0 → The equation has no real roots (the roots are imaginary).

Thus, the discriminant determines the nature of the roots of a quadratic equation.

Sum and Product of Roots of a Quadratic Equation

For a quadratic equation of the form ax² + bx + c = 0, the sum and product of the roots can be found directly using the coefficients of the equation without actually solving it.

Let the roots of the equation be α and β.

• Sum of the roots: α + β = −b/a
• Product of the roots: αβ = c/a

Example: Find the sum and the product of the roots of the equation 2x² + 5x + 3 = 0.
Solution:

Given quadratic equation, 2x² + 5x + 3  = 0

Comparing with, ax² + bx + c = 0

We get, a = 2, b = 5, c =3

• Sum of Roots = α + β = -b/a = -5/2
• Product of Roots = αβ = c/a = 3/2

Writing Quadratic Equations Using Roots

If α and β are the roots of a quadratic equation, the equation can be written using the formula x² − (α + β)x + αβ = 0

This means the sum of the roots (α + β) and the product of the roots (αβ) help in forming the quadratic equation.

• Another way to write the equation is (x − α)(x − β) = 0

Example: Find the quadratic equation whose roots are 4 and −1.

Solution:

Given, α = 4 and β = −1

Sum of roots: α + β = 4 + (−1) = 3

Product of roots: αβ = 4 × (−1) = −4

Using the formula: x² − (α + β)x + αβ = 0

x² − 3x − 4 = 0

Therefore, x² − 3x − 4 = 0 is the required quadratic equation.

Quadratic Equations Having Common Roots

Two quadratic equations are said to have common roots if they share at least one solution.

Consider two quadratic equations:

• a₁x² + b₁x + c₁ = 0
• a₂x² + b₂x + c₂ = 0

If these equations have one common root, then the following condition must be satisfied: (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁) = (a₂c₁ − a₁c₂)²

If both roots are common, then the coefficients of the equations are proportional:

a₁/a₂ = b₁/b₂ = c₁/c₂

Thus, two quadratic equations can have either one common root or both roots common, depending on the relationship between their coefficients.

Example: Find the value of k such that the quadratic equations x² − 5x + 6 = 0 and x² − 3x + k = 0 have a common root.

Solution:

Let the common root be α.

From the first equation:
α² − 5α + 6 = 0 ...(1)

From the second equation:
α² − 3α + k = 0 ...(2)

Subtract (2) from (1): (α² − 5α + 6) − (α² − 3α + k) = 0

−2α + 6 − k = 0

α = (6 − k)/2

Substitute in equation (1):

((6 − k)/2)² − 5((6 − k)/2) + 6 = 0

k² − 2k = 0

k(k − 2) = 0

k = 0 or k = 2

Therefore, the values of k are 0 and 2.

Methods for Quadratic Roots

Finding the roots of a quadratic equation means determining the values of x that satisfy the equation ax² + bx + c = 0. The points that satisfy this equation are called solutions or roots of this quadratic equation.

These are the four common methods for solving a quadratic equation: 

• Factorisationa Factorisation This method involves rewriting the quadratic equation as a product of two binomials. If the quadratic equation is factorable, it can be expressed in the form (px + q)(rx + s) = 0, where "q/p" and "s/r" will be the roots of the quadratic Equation.

• Completing the Square Method: This method involves rewriting the quadratic equation in the form of a perfect square trinomial. By adding and subtracting the appropriate value to both sides of the equation. which can be solved by taking the square root of both sides.

• Graphical Method: The graphical method of solving a quadratic equation involves plotting the corresponding quadratic function f(x) = ax² + bx + c on a graph and finding the points where the graph intersects the x-axis.

• Quadratic Formula: The quadratic formula is a general method that can be used to solve any quadratic equation.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a, b, and c are the coefficients from the quadratic equation ax + bx + c = 0.

Maximum and Minimum Value of a Quadratic Equation

A quadratic function is written as f(x) = ax² + bx + c. Its graph is a parabola. The maximum or minimum value of the function occurs at the vertex of the parabola, whose x-coordinate is x = −b / 2a

• If a > 0, the parabola opens upward and the function has a minimum value at x = −b/2a.
  Range: [f(−b/2a), ∞)
• If a < 0, the parabola opens downward, and the function has a maximum value at x = −b/2a.
  Range: (−∞, f(−b/2a)].

The domain of a quadratic function is all real numbers (−∞, ∞).

Solved Examples

Example 1: Check whether the following equation is a quadratic equation or not. (x - 2)(x + 1) = (x - 1)(x + 3) 
Solution:

We know that a quadratic equation must be of degree 2. 

Let's simplify and check the given equation. 

 (x - 2)(x + 1) = (x - 1)(x + 3)

⇒ x² + x - 2x - 2 = x² + 3x - x - 3
⇒ x² - x - 2 = x² + 2x - 3
⇒ -x - 2 = 2x - 3 
⇒ -3x + 1 = 0  

This equation is of degree 1. Thus, it cannot be a quadratic equation. 

Example 2: Find the quadratic equation having the roots 4 and 9, respectively.
Solution:

The quadratic equation having the roots α, β, is (x - α)(x - β) = 0

Given,
α = 4, and β = 9

Therefore the required quadratic equation is,
(x - 4)(x - 9) = 0
x² - 9x - 4x + 36 = 0
x² - 13x + 36 = 0

Thus, the required quadratic equation is x² - 13x + 36 = 0

Example 3: The equation 3x² + 5x + 9 = 0 has roots α and β. Find the quadratic equation whose roots are 1/α and 1/β.
Solution:

Given equation 3x² + 5x + 9 = 0

Comparing with ax² + bx + c = 0
a = 3, b = 5 and c = 9

α + β = -b/a = -5/3
αβ = c/a = 9/3 = 3

Roots of the new equation are 1/α and 1/β.
Sum of Roots = (α + β)/α β = (-5/3)/(3) = -5/9
x² - (-5/9)x + 1/3 = 0
Simplifying,

9x² + 5x + 3 = 0

Practice Questions

1. Find the sum and product of the roots of the quadratic equation: 4x² + 7x - 5 = 0.
2. Write the quadratic equation whose roots are -3 and 6.
3. Find the quadratic equation whose sum of roots is 5 and product of roots is 6.
4. Solve the quadratic equation x² - 6x + 8 = 0 by factoring.
5. Use the quadratic formula to solve the quadratic equation: 2x² + 3x - 2 = 0.