We will discuss here about some of the general properties of quadratic equation.
We know that the general form of quadratic equation is ax^2 + bx + c = 0, where a is the coefficient of x^2, b is the coefficient of x, c is the constant term and a ≠ 0, since, if a = 0, then the equation will no longer remain a quadratic
When we are expressing any quadratic equation in the form of ax^2 + bx + c =0, we have in the left side of the equation a quadratic expression.
For example, we can write the quadratic equation x^2 + 3x = 10 as x^2 + 3x – 10 = 0.
Now we will learn how to factorize the above quadratic expression.
x^2 + 3x  10
= x^2 + 5x  2x  10
= x(x + 5) 2 (x + 5)
= (x + 5)(x – 2),
Therefore, x^2 + 3x – 10 = (x + 5)(x – 2) ............ (A)
Note: We know that mn = 0 implies that, either (i) m = 0 or n = 0 or (ii) m = 0 and n = 0. It is not possible that both of m and n are nonzero.
From (A) we get,
(x + 5)(x – 2) = 0, then any one of x + 5 and x  2 must be zero.
So, factorizing the left side of the equation x^2 + 3x – 10 = 0 we get, (x + 5)(x – 2) = 0
Therefore, any one of (x + 5) and (x – 2) must be zero
i.e., x + 5 = 0 ................ (I)
or, x – 2 = 0 .................. (II)
Both of (I) and (II) represent linear equations, which we can solve to get the value of x.
From equation (I), we get x = 5 and from equation (II), we get x = 2.
Therefore the solutions of the equation are x = 5 and x = 2.
We will solve a quadratic equation in the following way:
(i) First we need to express the given equation in the general form of the quadratic equation ax^2 + bx + c = 0, then
(ii) We need to factorize the left side of the quadratic equation,
(iii) Now express each of the two factor equals to 0 and solve them
(iv)The two solutions are called the roots of the given quadratic equation.
Notes: (i) If b ≠ 0 and c = 0, one root of the quadratic equation is always zero.
For example, in the equation 2x^2  7x = 0, there is no constant term. Now factoring the left side of the equation, we get x(2x  7).
Therefore, x(2x  7) = 0.
Thus, either x = 0 or, 2x – 7 = 0
either x = 0 or, x = 7/2
Therefore, the two roots of the equation 2x^2  7x = 0 are 0, 7/2.
(ii) If b = 0, c = 0, both the roots of the quadratic equation will be zero. For example, if 11x^2 = 0, then dividing both sides by 11, we get x^2 = 0 or x = 0, 0.
Algebra/Linear Algebra
Introduction to Quadratic Equation
Formation of Quadratic Equation in One Variable
General Properties of Quadratic Equation
Methods of Solving Quadratic Equations
Examine the Roots of a Quadratic Equation
Problems on Quadratic Equations
Quadratic Equations by Factoring
Word Problems Using Quadratic Formula
Examples on Quadratic Equations
Word Problems on Quadratic Equations by Factoring
Worksheet on Formation of Quadratic Equation in One Variable
Worksheet on Quadratic Formula
Worksheet on Nature of the Roots of a Quadratic Equation
Worksheet on Word Problems on Quadratic Equations by Factoring
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