Introduction to Quadratic Equations and Finding the Roots
Introduction to Quadratic Equations and Finding the Roots: Quadratic equations are commonly used in everyday life, such as when calculating areas, a product’s profit, or estimating an object’s speed. It can be defined as an equation in one variable in which the maximum power of the variable is , with the most common form being .
Here, is the variable, with coefficients as , and , and the . A quadratic equation can have a maximum of roots, and these roots can be real or imaginary depending on the sign of the discriminant. Now, what is discriminant? Let us study the formula of discriminant, the nature of roots of the quadratic equation, and its graph in the following article.
What is the Degree of a Polynomial?
The degree of a polynomial is the highest power of a variable in a polynomial equation. For example: is a polynomial of degree .
We also classify the polynomial on the basis of degrees as follows:
Thus, a polynomial of degree is known as a Quadratic Polynomial. A quadratic polynomial, when equated to zero, becomes a quadratic equation.
Roots of a Quadratic Equation
The roots of a quadratic equation are the values of variables that satisfy that equation. In other words, if , then is known as a root of quadratic equation .
Relationship Between Roots and Degree of a Polynomial Equation
According to the fundamental theorem of algebra, the maximum number of roots a polynomial can have equals the degree of the polynomial. For example, the number of roots of a linear polynomial is , the maximum number of roots in a quadratic polynomial and cubic polynomial is and , respectively.
General Form of a Quadratic Equation
The general form of a quadratic equation is
Where, and are numbers (real or complex), and is the variable.
The following theorem suggests the number of roots of a quadratic equation.
Theorem: A quadratic equation cannot have more than two roots.
Proof: If possible, let be three distinct roots of the quadratic equation , where and .
Then, each one of will satisfy this equation.
Subtracting from , we get,
[ and are distinct ]
Subtracting from we get,
[ and are distinct ]
Subtracting from , we get,
But, this is not possible because and are distinct.
Thus, the assumption that a quadratic equation has three distinct real roots is wrong.
Hence, it is proved that a quadratic equation cannot have more than roots.
Remark: It follows from the above theorem that if a quadratic equation is satisfied by more than two values of , it is satisfied by every value of , and it is an identity.
Quadratic Equations with Real Coefficients
Consider the quadratic equation:
Where, and
Multiplying both sides of by , we get
Thus, the quadratic equation has two roots, say and given by
Where , and .
Discriminant of a Quadratic Equation
We observed that the roots of the equation are given by
where , and
Thus, the nature of the roots depends on the quantity under the square root sign, i.e. . This quantity is called the discriminant of the equation and is usually denoted by .
Nature of Roots of a Quadratic Equation
Case 1: When are real numbers, .
Condition
Nature of Roots
Real and Equal
Real and Distinct
Complex, a pair of complex conjugates
Case 2: When are rational numbers,
Condition
Value of
Nature of Roots
–
Rational and Equal
Perfect square of a rational number
Rational and Distinct
Not a perfect square of a rational number
Irrational and Distinct, a pair of irrational conjugates of the form: where
–
Complex, a pair of complex conjugates
Graph of Quadratic Function
Let
i.e.
represents a parabola with vertex at and faces upwards if and downwards if .
Case 1: When , the equation has two equal real roots , and which is a parabola touching axis.
Case 2: When the equation has no real roots, and , which represents a parabola with vertex at .
Case 3: When , the equation has two distinct real roots, say . Then represents a parabola with vertex at .
Sign of the Expression
The sign of for various values of , and are tabulated below.
No.
Value of
Value of
Value of
Sign of
1.
Equal to zero
Same as , except at , where it is zero
2.
3.
Less than zero
Same as
4.
5.
More than zero
Opposite sign to when
Same sign otherwise
Maxima and Minima
From the graphs, we see that maximum or minimum occurs at , and its corresponding value is . When , we have a minima, and when , we have maxima.
Quadratic Inequalities
A quadratic inequality is an equation of second degree that uses an inequality sign instead of an equal sign. Example:
Steps to Solve Quadratic Inequalities
Step 1: Consider the quadratic inequality , i.e. , where .
Step 2: Make linear factors in L.H.S. using the method of factorisation.
Step 3: Let the roots of the equation be and . So, write the given inequation as , where .
Step 4: Thus, the critical points corresponding to the given inequation are and . Plot these critical points on a real number line
Step 5: Divide the number line into three sub intervals and such that on or , and on .
Note: The values of for the right most interval on the above number line is positive, and then for the remaining intervals, the sign of changes alternatively.
Step 6: The required solution for is .
Note:
If we have inequality sign as then the solution for this inequality will be .
Similarly, the solutions for is , and the solution for is .
Relationship Between Roots and Coefficients of a Quadratic Equation
If and are roots of a quadratic equation, , where then, we have
Sum of roots
Product of roots
Formation of Quadratic Equation From Roots
Let and be the two roots of a quadratic equation .
Now, .
Using , and , we have
Hence, if and be the roots of the quadratic equation, then the quadratic equation is given by
Solved Examples – Introduction to Quadratic Equations and Finding the Roots
Q.1. Discuss the sign of the following quadratic function: Ans: Given: Comparing it with , we have Substituting the known values we get, Thus, Hence, for all real . It carries a positive sign.
Q.2. Solve the following equation Ans: Given: Comparing it with , we get Discriminant
Q.3. Find the roots of the equation: Ans: Given: Comparing it with , we get Hence, the roots of the given equation are and .
Q.4. Solve for : . Ans:Step 1: Factorise the L.H.S., Step 2: Determine the critical values of . Here, the critical values are . Step 3: Plotting these critical values on the real number line, we have
Step 4: Let , we need to check for the sign of in the intervals, and . Thus, we have
Step 5: We observe that for . Hence, the required solution is
Q.5. Determine a positive real value of such that both the equations and may have real roots. Ans: The given equations are and As both the equations have real roots, the discriminant of each . Thus, we have and and and or and But is a positive real number. So, we have and Hence, the required value of is .
Summary of Introduction to Quadratic Equations and Finding the Roots
A quadratic equation is defined as an equation in one variable in which the maximum power of the variable is . The roots of the quadratic equation are , where . If a quadratic equation with real coefficients has discriminant , then the roots are real and equal; if , then the roots are real and repeated; if , then the roots are complex. Also, the sign of the quadratic function changes with the values of and . If and be the roots of the quadratic equation, then the quadratic equation is given by . Similarly, for , where , we know that .
Frequently Asked Questions (FAQs)
Q.1. How do you find the roots of a quadratic equation? Ans: The roots of the quadratic equation , where , are given by
Q.2. How do you introduce quadratic equations? Ans: A quadratic equation is defined as an equation in one variable in which the maximum power of the variable is . Examples of quadratic equations are , and .
Q.3. What are quadratic roots? Ans: The values of a variable that satisfy the given quadratic equation are the roots of that quadratic equation or quadratic roots.
Q.4. How do you write a quadratic equation with given roots and points? Ans: If and be the roots of the quadratic equation, then the quadratic equation is given by If the roots of a quadratic equation are given, then we can determine the quadratic equation, using the formula,
Q.5. How do you find if a quadratic equation has real roots? Ans: The discriminant of a quadratic equation is given by If then the quadratic equation has real and unequal roots, and if , then the quadratic equation has real and equal roots. Thus the roots of a quadratic equation are real for .
We hope this detailed article on Introduction to Quadratic Equations and Finding the Roots will make you familiar with the topic. If you have any inquiries, feel to post them in the comment box. Stay tuned to embibe.com for more information.