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 co-efficient 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 non-zero.
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.`