Types of Quadratic Equations and their Significance in the CAT Exam

Quadratic equations are not the same as the linear equation. The fundamental distinction between a quadratic equation and a linear equation is that the latter will have an x2 term. A term is considered quadratic or a term with a degree 2 when the power of the term is 2. Let’s find out more about it.

Basic Concepts

Some of the key concepts for addressing a quadratic problem are as follows. A quadratic equation can be solved in two ways:

1. Using a standard formula.

2. The method of factorisation.

1. Using a Standard Formula

This approach can be used to solve any quadratic problem quickly and easily. The solution will be ax2 + bx + c = 0 if the quadratic equation is of the type ax2 + bx + c = 0.

x = -b ± √(b² -4ac)/2a.

We will acquire two types of numbers using this method: one using the + sign and the other using the – sign. The formula can be applied to any type of quadratic equation, regardless of whether or not it can be factored. Let’s look at an example to better grasp the procedure.

Q. x² – 2x – 6 = 0

The equation you’ve been given is in standard form. When you compare the numbers, you’ll notice that a = 1, b = -2, and c = -6. These values will be entered into the usual equation described above. We’re going to get:

x = -(-2) ± √(-2) 2 – 4(1)(-6)/ 2(1)

x = 2 ±√4 + 24/2

x = 2 ± √28/2

x = 2 ± 2√7/2

x = 1 ± √7

1. Standard Form: y=ax2+bx+cy=ax2+bx+c.

2. Factored Form: y=a(x-r1)(x-r2)y=a(x−r1)(x−r2).

3. Vertex Form: y=a(x-h)2+ky=a(x−h)2+k.

Each quadratic form has a distinct appearance, allowing different problems to be solved more easily in one form than another. We’ll go over the differences between each form and how to transition between them.

Formula for a Quadratic Equation

The quadratic formula gives the solution or roots of a quadratic equation:

(α, β) = [-b ± √(b2 – 4ac)]/2a

How to Solve Quadratic Equations

A quadratic equation can be solved in two ways.

• Algebraic Approach

• Graphical Approach

Solving Quadratic Equations Using the Algebraic Method

ax2 + bx + c = 0 in general form;

x2 + bx/a + c/a = 0

⇒ (x + b/2a)2 = b2/4a2 – c/a

Or, (x + b/2a)2 = (b2 – 4ac)/4a2

Or, x + b/2a = ± (√b2 – 4ac)/2a

⇒ x = [-b ± √(b2 – 4ac)]/2a

b2 – 4ac = Discriminant (D)

• α = (-b+√D)/2a

• β = (-b – √D)/2a

α+β= -b/a, α.β = c/a

As a result, the quadratic equation can be expressed as follows:

⇒ x2 – (α + β)x + (α.β) = 0.

Quadratic Equation Graphical Solution

Consider the quadratic equation ax2 + bx + c = 0, in which a, b, and c are real numbers and a is less than zero.The following is a rewrite of the expression:

a[(x + b/2a)2 + (D/4a2)]

The parabola with the vertex at P [-b/2a, -D/4a] and axis parallel to the y-axis is represented by the quadratic equation above.

The value of ‘a’ in a quadratic equation affects whether the graph of the quadratic equation will be concave upwards (a > 0) or downwards (a 0). The discriminant (b2 – 4ac) value influences whether a quadratic equation’s graph will:

• • Make two x-axis intersections. b2 – 4ac > 0 is an example of this
• • Only comes close to touching the x-axis, i.e. b2 – 4ac = 0
• • Does not cross the x-axis, i.e. b2 – 4ac 0

Conclusion

In mathematics, almost every student encounters the quadratic formula, which is a popular method for determining the roots of a quadratic problem. The quadratic formula is useful in determining the area of a space, the speed of a moving object, the value of profit gained on a product, and other things in real life.