Quadratic Equations

A Quadratic equation is a second-degree polynomial equation that can be represented as ax² + bx + c = 0. In this equation, x is an unknown variable, a, b, and c are constants, and a is not equal to 0. The solutions of a quadratic equation are known as its roots. These roots can be found using methods like factoring, completing the square, using the quadratic formula, or graphing.

Example:

• x² - 11x + 28
• 3x² - 11x + 23 = 0

Standard Form of Quadratic Equation

The standard form of an equation is the conventional or widely accepted way of writing equations that simplifies their interpretation and makes it easier for calculations. Standard Form of Quadratic Equation is:

ax² + bx + c = 0

• x is Variable of Equation
• a, b, and c are Real Numbers and Constants and a ≠ 0

In general, any second-degree polynomial P(x), in form of P(x) = 0 represents a Quadratic Equation.

Quadratic equation contains a variable raised to the power of 2 (squared) as its highest exponent.

Roots of Quadratic Equation

The Roots of Quadratic equation are the two values of x, which satisfies the condition q(x)=0. This implies that for any x_o if q(x) = 0. Then x_o is the root of the q(x).

In other words, the roots are the solutions to the equation, or the points where the graph of the quadratic function (a parabola) intersects the x-axis.
The roots are also known as zeros of Quadratic Equation.

Example, For a quadratic equation q(x): 3x² - 10x - 8 = 0 the roots are x = -2/3 and x = 4.

As for x = -2/3, 
q(-2/3) = 3(-2/3)² - 10(-2/3) - 8 
⇒ q(-2/3)  = 4/3 + 20/3 - 8 
⇒ q(-2/3) = 0

Note: Quadratic equation is a two degrees polynomial i.e., it can have a maximum of 2 roots.

Nature of Roots of Quadratic Equation

A Quadratic Equation can have "Real or Imaginary" roots that can be determined by the value of discriminant(D) of a Quadratic Equation.

Discriminant of a quadratic equation : D = b² - 4ac

Positive Coefficient (a > 0):

• If b and c are both positive, the polynomial will have no positive roots.
• If b and c are both negative, the polynomial will have two positive roots.
• If b and c have opposite signs, the polynomial will have one positive root.

Negative Coefficient (a < 0):

• If b and c are both positive, the polynomial will have two negative roots.
• If b and c are both negative, the polynomial will have no negative roots.
• If b and c have opposite signs, the polynomial will have one negative root.

Sum of Roots in Quadratic Equation

For a given quadratic equation ax² + bx + c = 0, the sum of roots can be found with the help of the coefficient of x², and coefficient of x.

Formula for Sum of roots of Quadratic Equation :

Sum of Roots: α + β = -b/a = -Coefficient of x/ Coefficient of x²

Product of Roots in Quadratic Equation

For a quadratic equation of the form, ax² + bx + c = 0, the product of roots can be found with the help of the coefficient of x², and the constant term.

Formula for Product of Roots of Quadratic Equation:

Product of Roots: αβ = c/a = Constant term/ Coefficient of x²

Let's try to understand the formulas for the sum and product of roots with the help of an example.

Example: Find the sum and the product of the roots of 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 the roots of the quadratic equation α, and β are given then we can easily write the equation by using the formula,

(x - α)(x - β) = 0

• If the sum (α + β) and the product (αβ) of the quadratic equation then the quadratic equation is given using x² - (α + β)x + αβ = 0.

Let's understand this with the help of an example.

Example 1: Find the quadratic equation whose roots are, 1 and 2.
Solution:

Given, α = 1 and β = 2

Then,
(x - α)(x - β) = 0

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

x² - 3x + 2 = 0 is the required quadratic equation.

We can also find the quadratic equation if the sum and the product of its roots are given. Let us suppose the sum (S) and the product (P) of the quadratic equation are (α + β) and αβ respectively.

Formula to write Quadratic Equation when Sum and Product of Roots are given :

x² - (Sum)x + (Product) = 0
or, x² - (α + β)x + αβ = 0

Example 2: Find the quadratic equation whose sum of the roots is, 3 and the product of the root are 2.

Solution:

Given, α + β = 3 and αβ = 2

Then the required quadratic equation is,

x² - (α + β)x + αβ = 0
⇒ x² - 3x + 2 = 0

x² - 3x + 2 = 0 is the required quadratic equation.

Methods to find the root a quadratic equation

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

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

• Factorization Method : 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 ax²+ bx + c = 0.

Maximum and Minimum Value of Quadratic Equation

We know that we can easily plot a graph of the quadratic equation in the form of q(x) = ax² + bx + c and it comes out to be a parabola.

Now, performing the necessary calculation we can state that:

• If a > 0, q(x) is minimum at x = -b/2a
• If a < 0, q(x) is maximum at x = -b/2a

Thus we can easily get the range of the quadratic equation.

• When a > 0, Range of q(x) is [q(-b/2a), ∞)
• When a < 0, Range of q(x) is (-∞, q(-b/2a)]

Maximum and Minimum values of the function is found by studying the cases added below:

Case 1: If a > 0 in ax² + bx + c = 0,

• Then range of quadratic equation is [-D/4a, ∞)

Case 2: If a > 0 in ax² + bx + c = 0,

• Then range of quadratic equation is (-∞, -D/4a)

We can also find the maximum and minimum value of the function in the given interval by using concept of Maxima and Minima.

Solved Examples on Quadratic Equation

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: Equation 3x² + 5x + 9 = 0 has roots α, and β. Find the quadratic equation having the roots 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 = 1/α + 1/β = (α + β)/α β = (-5/3)×(1/3) = -5/9
Product of Roots = 1/α β = 1/3

Thus, the required quadratic equation is,

x² - (Sum)x + Product = 0
x² - (-5/9)x + 1/3 = 0
Simplifying,

9x² + 5x + 3 = 0

Practice Questions

Solve these practice questions on quadratic equations.

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.
6. Find the roots of the quadratic equation 3x² - 4x + 1 = 0 using the discriminant method.
7. Solve the quadratic equation 5x² - 20x + 25 = 0 by completing the square.
8. Determine whether the roots of the equation 6x² + 11x + 3 = 0 are real or imaginary using the discriminant.
9. If the roots of a quadratic equation are 2 and 4, write the equation.
10. Find the maximum or minimum value of the quadratic function f(x) = -3x² + 12x - 5.

Read More,

• Factorization of Polynomials 
• Find the Equation of a Quadratic Function
• Quadratic Equation Class 10