In this section, we will examine the roots of a quadratic equation.
That is, we will analyse whether the roots of a quadratic equation are equal or unequal, real or imaginary and rational or irrational.
To examine the roots of a quadratic equation, let us consider the general form a quadratic equation.
ax2 + bx + c = 0
(Here a, b and c are real and rational numbers)
To know the nature of the roots of a quadratic-equation, we will be using the discriminant b2 - 4ac.
Because b2 - 4ac discriminates the nature of the roots.
Let us see how this discriminant b2 - 4ac can be used to know the nature of the roots of a quadratic-equation.
Example 1 :
Examine the nature of the roots of the following quadratic equation.
3x2 + 8x + 4 = 0
Solution :
The given quadratic equation is in the general form
ax2 + bx + c = 0
Then, we have a = 3, b = 8 and c = 4.
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = 82 - 4(3)(4)
b2 - 4ac = 64 - 48
b2 - 4ac = 16
Here, b2 - 4ac > 0 and also a perfect square.
So, the roots are real, unequal and rational.
Example 2 :
Examine the nature of the roots of the following quadratic equation.
2x2 - 3x - 1 = 0
Solution :
The given quadratic equation is in the general form
ax2 + bx + c = 0
Then, we have a = 2, b = -3 and c = -1.
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-3)2 - 4(2)(-1)
b2 - 4ac = 9 + 8
b2 - 4ac = 17
Here, b2 - 4ac > 0, but not a perfect square.
So, the roots are real, unequal and irrational.
Example 3 :
Examine the nature of the roots of the following quadratic equation.
x2 - 16x + 64 = 0
Solution :
The given quadratic equation is in the general form
ax2 + bx + c = 0
Then, we have a = 1, b = -16 and c = 64.
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-16)2 - 4(1)(64)
b2 - 4ac = 256 - 256
b2 - 4ac = 0
So, the roots are real, equal and rational.
Example 4 :
Examine the nature of the roots of the following quadratic equation.
5x2 - 4x + 2 = 0
Solution :
The given quadratic equation is in the general form
ax2 + bx + c = 0
Then, we have a = 5, b = -4 and c = 2.
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-4)2 - 4(5)(2)
b2 - 4ac = 16 - 40
b2 - 4ac = -24
Here, b2 - 4ac < 0.
So, the roots are imaginary.
Problem 5 :
Examine the nature of the roots of the following quadratic equation.
1/(x+1) + 2/(x-4) = 2
Solution :
The given quadratic equation is not in the general form.
First, write the given quadratic equation in the general form.
1/(x+1) + 2/(x-4) = 2
Add the two fractions on the right side of the equation using cross multiplication.
[(x-4) + 2(x+1)] / [(x+1)(x-4)] = 2
(x - 4 + 2x + 2) / (x2 - 3x - 4) = 2
(3x - 2) / (x2 - 3x - 4) = 2
Multiply each side by (x2 - 3x - 4).
3x - 2 = 2(x2 - 3x - 4)
3x - 2 = 2x2 - 6x - 8
2x2 - 9x - 6 = 0
Now, the quadratic equation is the general form
ax2 + bx + c = 0
Then, we have a = 2, b = -9 and c = -6.
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-9)2 - 4(2)(-6)
b2 - 4ac = 81 + 48
b2 - 4ac = 129
Here, b2 - 4ac > 0, but not a perfect square.
So, the roots are real, unequal and irrational.
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