Quadratic Equation is one of the most popular topic in some of competitive exams. Normally, questions asked from this section are based on simple concepts only. Besides some tricks of this concept, you just need to remember 3-4 points which will help you to solve any kind of questions asked.
If `ax^2 + bx + c = 0` be a quadratic equation and `alpha` , `beta` are the roots,
Quadratic Equation
Standard form of quadratic equation:
`ax^2 + bx + c = 0`
where, a, b and c are real numbers.
For example, `4x^2 + 2x + 10 = 0` is a quadratic equation ( Degree= 2)
Note: Degree = Highest power in the equation.
Roots of Quadratic Equations
Roots are the values (of x) which satisfies the given equation.
For example: If equation (`x^2 + 4x - 12 = 0`) has roots 2 and -6. Both will satisfies the equation. Let us see how:
If x = `2`,
`2^2 + 4(2) - 12 = 12- 12= 0`
Therefore, x=`2` satisfies the equation
Similarly, x= `-6` will satisfy the equation.
Example1: Which of the following is one of the roots of the equation:
`x^2 + 5x - 6= 0`
a) 5 b) 1 c)3 d) 6
Solution: The easiest way to solve this question is go through the options. Put the values in the equation and check whether it satisfies or not.
If x= `5`
`5^2 + 5(5) - 6= 44 ne 0`
Therefore, `5` is not the answer
If x= `1`
`1^2 + 5 -6` = `0`
It satisfies the equation. Therefore `1` is answer
Basic terms of Quadratic Equation
Now, as I told you quadratic equation has two roots. These two roots has some definite relatin with the real values of the equations. Let me tell you how:
Let `ax^2 + bx + c = 0` be a quadratic equation and `alpha` , `beta` are the roots, then
`alpha` + `beta` = `-b/a` i.e.sum of roots and
`alpha` `beta` = `c/a` i.e. product of roots
Remember this point, because it will be helpful to solve various questions.
Summarizing,
If `ax^2 + bx + c = 0` be a quadratic equation and `alpha` , `beta` are the roots,
- Number of roots = 2 i.e. degree of equation.
- Both roots will satisfy the equation.
- `alpha` + `beta` = `-b/a` and `alpha` `beta` = `c/a`
Example2: What is the sum of the roots of following equation:
`x^2 + 2x - 3 = 0`?
Solution: Sum of roots = `alpha` + ` beta` = `-b/a`
⇒Sum = `-2/1` = `-2` Ans
Example3: What is the product of roots of the same equation: `x^2 + 2x - 3 = 0`?
Solution: Product of roots = `alpha` `beta` = `c/a`
⇒ Product= `-3/1` = `-3` Ans
Factorization of Quadratic Equation
Consider the following equation:
`x^2 + 5x - 6= 0`
⇒ (x-1) (x+6)= 0
Factors of equation: (x-1) (x+6)
Therefore, roots = 1, -6
Now, I will tell you the method of finding the factors of equation:
- Start with constant term
- Factorize the constant term
- After factorizing, choose that pair whose: addition or subtraction gives coefficient of x
Groups of 6 = {6,1}, {2,3}
Now, we will choose that pair, whose addition or subtraction gives `5`
⇒{6,1}
Equation becomes: `x^2 + (6x - 1x) - 6= 0`
⇒`x^2 + 6x - 1x- 6= 0`
⇒`x (x+6) -1 (x+6) = 0`
⇒`(x-1) (x+6) =0`
Therefore, (x-1) and (x+6) are the factors of given equation.
Example4: Find the factors of Quadratic equation:
`x^2 + 4x + 4 = 0`
Solution: `x^2 + 4x + 4 = 0`
⇒`x^2 + (2x+2x) + 4 =0`
⇒x (x+2) + 2 (x+2) = 0
⇒(x+2) (x+2) = 0
Therefore, (x+2) (x+2) are the factors.
Example5: What are the roots of following equation:
`x^2 - x = 0`
Solution: `x^2 - x = 0`
⇒ x (x-1) = 0
⇒ (x+0) (x-1) are the factors
Roots = 0 and 1
I hope you understand this concept. For your better understanding, I will soon update the video lesson of this chapter.
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