Today our topic of discussion is quadratic equations. The topic is very common in all the competitive exams and requires a lot of practice. It is very different than the linear equation. The main difference between the quadratic equation and linear equation is that the latter will have a term x² in it. When the power of a term is 2 it is called quadratic term or a term with a degree 2. Let us learn more about it.
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Quadratic Equation
A quadratic equation is usually defined as ax² + bx + c = 0, where a, b, and care the real terms and a is not equal to 0. Because if a = 0, then the equation will be linear.
Basic Concepts
Here are some of the basic concepts to solve a quadratic equation. To solve a quadratic equation you can use two methods:
1. Using a standard formula
2. Factorization method
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1. Using a Standard Formula
Any quadratic equation can be solved easily and quickly by using this method. If the quadratic equation is of the type ax² + bx + c = 0, then the solution will be
x = -b ± √(b² -4ac)/2a.
In this method, we will get two types of values, one will be due to + sign and other will be with the use of – sign. The formula can be used for all types of quadratic equation irrespective of whether the equations can be factorized or not. Let us look into an example to understand the method better.
Q. x² – 2x – 6 = 0
Here the equation given to you is in the standard form. On comparing you can find that a = 1, b = -2, c = -6. We will put these values in the standard equation given above. We will 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
Thus, we can find two values of the given equation, (1 + √7) and (1 – √7). As the roots contain √7 which is an irrational number, the given values are irrational.
2. Factorization Method
In factorization, you need to find the factors of the given terms. Unlike the above method, this method can only be solved when the equation given to us can be factorized. For example, factors of 8 are 1, 2, 4, and 8 Similarly factors of 12 are 1, 2, 3, 4, 6, and 12. Thus you can use this method only when the terms given to you can be factorized. This method is usually quicker than the other method.
Q. Solve x² -5x + 6 = 0
The given equation is in the standard quadratic form. To solve it using factorization method you need to look for two terms i.e. b and c. Here, b = -5 and c = 6. You have to check the product of which two numbers will be 6 and some of which two numbers will be -5. For this find the factors of 6, which are 1, 2, 3, and 6. Thus, you can see that product of 2 and 3 will be 6 and sum of 2 and 3 will be 5. But we need to find -5, therefore we need to take -2 and -3 whose sum will be -5 and product will be 6.
So, the factors for the given equation are (x + 2) and (x + 3).
The other basic concepts you need to remember while solving quadratic equations are:
1. Nature of roots
2. Sum and product of the roots
3. Forming a quadratic equation
1. Nature of the Roots
Nature of roots is necessary to determine whether the given roots of the equation are real, imaginary, rational or irrational. The basic formula you need to remember is b² – 4ac. This formula is also called discriminant or D. Thus, D = b² – 4ac
Now, the nature of the roots depends on the value of D. Here are the necessary conditions to determine the nature of the roots.
If D < 0 => than the given roots are imaginary.
For D = 0 => the roots given are real and equal.
If D > 0 => the roots are real and unequal.
Further, for D > 0, if the equation is a perfect square than the given roots are rational, otherwise they are irrational.
2. Sum and Product of Roots
For all the equation and formulas you need to remember the standard form of a quadratic equation and see the terms based on it. For any given equation the sum of the roots will always be -b/a and the product of the roots will be c/a. Sum of roots is also defined as Α(alpha) + Β(beta), while the product of the roots is Α*Β. Thus, the standard quadratic equation can also be written as x2 – (Α + Β)x + Α*Β = 0.
3. Forming a Quadratic Equation
There are two cases when the equation can be formed.
1. When the roots of the equation are given.
2. When the product and some of the roots are given.
In the former, the equation can be easily formed by using the multiplication method. Whatever the roots are given to you add them to x with an opposite sign and multiply them to find the equation. In the latter, to form an equation just follow the above formula used in sum and product rule. Put the values given to you in the equation and you will find your equation.
Practice Questions
1. Find the nature of the roots of the given equation: x² + 2x + 3 = 0.
A. Rational and equal
B. Equal
C. Imaginary
D. Irrational
The correct answer is C.
2. If the roots of the equation are -3 and -8, then which of the following is that equation?
A. x² + 11x + 24 = 0
B. x² – 11x – 24 = 0
C. x² – 11x + 24 = 0
D. x² + 11x – 24 = 0
The correct answer is A.
3. What will be the sum of the roots of the equation 35x² – 2x + 1= 0
A. 1/35
B. 69/35
C. 2/35
D. -1/35
The correct answer is C.
4. Solve the equation x² – 7x + 10 = 0 and find its roots.
A. (3 +- √23)/2
B. (3 +- √-5)/2
C. (-8,2)
D. (8,-2)
The correct answer is B.
5. If alpha and beta are the roots of the equation x² – 9x + 14 = 0 then find the value of A² +B².
A. 25
B. 28
C. 53
D. 81
The correct answer is C.
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