Nature of Roots of Quadratic Equation

Introduction

• The solutions of the quadratic equation ax²+bx+c=0 serve as the quadratic equation's roots.

• They are the values of the variable x that the equation requires.

• The x-coordinates of the x-intercepts of a quadratic function are the roots of the function.

• A quadratic equation can only have a maximum of two roots because its degree is 2.

• The nature of these roots of the quadratic equation is solely dependent on the value of the determining factor which is called a discriminant of the quadratic equation about which we will be discussing in this tutorial.

Quadratic Equation

• A quadratic equation in algebra is any equation that can be transformed in standard form like the following −

  ax²+bx+c=0

• Where x stands for an unknown and a, b, and c are known numbers, where a≠0, is used.

• The equation is linear, not quadratic, if a = 0, as the ax^2 The term is absent.

• The coefficients of the equation, denoted by the numbers a, b, and c, are the quadratic coefficient, linear coefficient, and constant or free term, respectively.

Example:

x²+x+1=0 ,5x²-2=0,x²=0 are quadratic equations. (Because its highest degree is 2)

Quadratic Formula

• If ax²+bx+c=0 is the given quadratic equation then the quadratic formula to find the roots of this quadratic equation is given by;

  $$\mathrm{x=\frac{-b±\sqrt{b^2-4ac}}{2a}.}$$

• Each of these two answers is also referred to as a quadratic equation root (or zero).

• The x-values at which every parabola, specifically stated as ax²+bx+c=y crosses the x-axis are represented by these roots geometrically.

• The quadratic formula can be used to determine the zeros of any parabola as well as the axis of symmetry of the parabola and the total number of real zeros in the quadratic equation.

Discriminant

• The discriminant of a polynomial in mathematics is a number that depends on the coefficients and establishes different characteristics of the roots.

• It is typically described as a polynomial function of the original polynomial's coefficients.

• In number theory, algebraic geometry, and polynomial factoring, the discriminant is frequently used. The sign $\mathrm{\Delta}$ is frequently used to indicate it.

The quadratic polynomial ax²+bx+c with a≠0 has the following discriminant −

$$\mathrm{\Delta =b^2-4ac}$$

Nature of roots depending on value of discriminant

Δ=b²-4ac is the discriminant value of the quadratic equation ax²+bx+c=0

• If Δ=0 then the two roots of the quadratic equation are real and equal.

• If Δ>0 then the two roots of the quadratic equation are real and unequal.

• If Δ < 0 then the two roots of the quadratic equation are non-real.

Solved Examples

1. Is the equation x-2x²=0 a quadratic equation?

Solution:

A quadratic equation in algebra is any equation that can be transformed in standard form like the form ax²+bx+c=0

As -2x²+x+0=0 is of the form ax²+bx+c=0, hence the given equation is the quadratic equation.

2. What are the coefficients of the quadratic polynomial x²-3x+7?

Solution:

A quadratic polynomial in algebra is any polynomial that can be transformed in standard form like the form ax²+bx+c. since the given polynomial x²-3x+7 is of the form ax²+bx+c, hence the coefficients are a=1,b=-3,c=7

3. Determine the value of the discriminant of the following quadratic equation x²-8x=0

Solution

Δ=b²-4ac is the discriminant value of the quadratic equation ax²+bx+c=0

Thus the value of the discriminant of the given equation x²-8x=0 is given by;

$$\mathrm{\Delta=(-8)^2-4(1)(0)=64-0=64}$$

Thus the value of the discriminant for the given equation is 64.

4. What is the nature of roots of the quadratic equation 2x²-4x+4=0?

Solution:

Δ=b²-4ac is the discriminant value of the quadratic equation ax²+bx+c=0 and the nature of roots of the quadratic equation is determined by the value of this discriminant. We first the value of the discriminant here as follows −

$$\mathrm{\Delta=b^2-4ac=(-4)^2-4(2)(4)=16-32=-16< 0}$$

If Δ < 0 then the two roots of the quadratic equation are non-real. Therefore the roots of the quadratic equation are non real values which we are also called as complex numbers. That is the roots of the given equation are complex values.

5.Calculate the zeroes of the quadratic equation x²-7x=0 by the method of factorization and hence mention the nature of roots of this quadratic equation.

Solution:

$$\mathrm{x^2-7x=0}$$

$$\mathrm{x(x-7)=0}$$

$$\mathrm{x=0,x=7}$$

Since the zeroes, 0 & 7 are real and distinct numbers. Thus the nature of roots of the given quadratic equation is real and distinct.

Conclusion

In this article we learned about quadratic equations and their roots, the discriminant method to find the roots of a quadratic equation and how the value of discriminant helps determine the nature of the roots of the quadratic equation.

FAQs

1.What three components of quadratic equations are crucial?

The "a", "b", and "c" in "ax²+bx+c," which are just numbers and are the "numerical coefficients" of the quadratic equation ax²+bx+c=0, they've given you to solve, and are used in the quadratic formula. Hence they are crucial components of the quadratic equation ax²+bx+c=0

2.Which four methods are used to solve quadratic equations?

Factoring, using square roots, completing the square, and the quadratic formula are the four ways to solve a quadratic problem.

3.Why is an equation a quadratic called a quadratic?

A quadratic in mathematics is a particular kind of problem that involves squaring, or multiplying a variable by itself. This terminology comes from the fact that a square's area equals the product of its side length and itself. Quadratum, the Latin word meaning square, is where the word "quadratic" originates.

4.What determines how roots are formed?

The characteristics of a quadratic equation's roots are determined by the discriminant. The term "nature" refers to the different kinds of roots that can exist, including real, rational, irrational, and imaginary numbers.

5.How can you tell whether two roots are equal or not?

The discriminant, which is equal to b²-4ac, must be calculated in order to understand the type of roots of quadratic equations (of the form ax²+bx+c=0. The roots are uneven and real when the discriminant is bigger than zero. The roots are equal and real when discriminant equals zero.

6.May any parabola have no roots?

A quadratic function has no actual roots and the parabola it represents does not intersect the x-axis if its discriminant is less than zero. Thus a parabola can have no roots if it does not cut or touch the x-axis.

7.How do you know if a root is rational or irrational?

The roots are rational or irrational if the discriminant is positive and a perfect square. The roots are irrational if the discriminant is positive and not a perfect square.