A quadratic equation is a form of polynomial equation in a single variable, typically denoted as "x," and can be presented in the standard form: ax² + bx + c = 0. In this representation, "a," "b," and "c" are constants, with the condition that "a" must not equal 0.
1. Factorization
It is very simple method to to solve quadratic equations. Factorization give 2 linear equations
For example: x^{2} + 3x – 4 = 0
Here, a = 1, b = 3 and c = 4
Now, find two numbers whose product is – 4 and sum is 3.
So, the numbers are 4 and 1.
Therefore, two factors will be 4 and 1
2. Completing the Square Method
Every Quadratic question has always a square term. If we could get two square terms of quality sign we can get a linear equations. Middle term is called as ‘b’ and splited by (b/2)^{2}
For exampl : – x²+ 4x +4
Here x² =1, b= 4
(x+2)² = 3
Take the square of both side
Therefore, x = 0.27 or 3.73
Other basic concepts to remember while solving quadratic equations are:
1.Nature of roots
2. Sum and product of the roots
3. Forming a quadratic equation
Q1: Which of the following is the general form of a quadratic equation?
(a) x² + 3x – 2 = 0
(b) 3x + 2 = 5
(c) 2x – 4 = 0
(d) 5x² – 7x = 3
Ans: (a)
The general form of a quadratic equation is ax² + bx + c = 0, where “a,” “b,” and “c” are constants. Option A follows this format, representing a quadratic equation.
Q2: Which method is most suitable for solving the quadratic equation x² – 6x + 9 = 0?
(a) Factoring
(b) Quadratic Formula
(c) Completing the Square
(d) Graphical Analysis
Ans: (c)
The quadratic equation x² – 6x + 9 = 0 is a perfect square trinomial, and completing the square is the most efficient method to solve it. This method involves transforming the equation into a squared binomial, which makes it easy to find the solutions.
Q3: Which of the following quadratic equations has complex roots?
(a) 2x² + 4x + 5 = 0
(b) x² + 6x + 9 = 0
(c) 3x² – 6x + 3 = 0
(d) 5x² – 10x + 5 = 0
Ans: (a)
The discriminant (Δ) of a quadratic equation determines the nature of its roots. If Δ < 0, the equation has complex roots. For option A, the discriminant is 4² – 4(2)(5) = 16 – 40 = 24, which is negative. Therefore, the quadratic equation 2x² + 4x + 5 = 0 has complex roots.
Q4: What is the discriminant of the quadratic equation 2x² – 5x + 3 = 0?
(a) 1
(b) 11
(c) 19
(d) 11
Ans: (c)
The discriminant (Δ) of a quadratic equation ax² + bx + c = 0 is given by the expression Δ = b² – 4ac. Substituting the coefficients from the given equation, we have Δ = (5)² – 4(2)(3) = 25 – 24 = 1. Therefore, the discriminant is 1.
Q5: For the quadratic equation 2x² + 5x – 3 = 0, what are the roots?
(a) x = 3 and x = 1/2
(b) x = 3 and x = 1/2
(c) x = 3 and x = 1/2
(d) x = 3 and x = 1/2
Ans: (a)
The quadratic equation can be factored as (2x – 3)(x + 1) = 0. Setting each factor equal to zero gives 2x – 3 = 0 and x + 1 = 0. Solving for “x” in each equation yields x = 3/2 and x = 1. Therefore, the roots are x = 3 and x = 1/2.
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