Examining the roots of a quadratic equation means to see the type of its roots i.e., whether they are real or imaginary, rational or irrational, equal or unequal.
The nature of the roots of a quadratic equation depends entirely on the value of its discriminant b\(^{2}\) - 4ac.
In a quadratic equation ax\(^{2}\) + bx + c = 0, a ≠ 0 the coefficients a, b and c are real. We know, the roots (solution) of the equation ax\(^{2}\) + bx + c = 0 are given by x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).
1. If b\(^{2}\) - 4ac = 0 then the roots will be x = \(\frac{-b ± 0}{2a}\) = \(\frac{-b - 0}{2a}\), \(\frac{-b + 0}{2a}\) = \(\frac{-b}{2a}\), \(\frac{-b}{2a}\).
Clearly, \(\frac{-b}{2a}\) is a real number because b and a are real.
Thus, the roots of the equation ax\(^{2}\) + bx + c = 0 are real and equal if b\(^{2}\) – 4ac = 0.
2. If b\(^{2}\) - 4ac > 0 then \(\sqrt{b^{2} - 4ac}\) will be
real and non-zero. As a result, the roots of the equation ax\(^{2}\) + bx + c = 0
will be real and unequal (distinct) if b\(^{2}\) - 4ac > 0.
3. If b\(^{2}\) - 4ac < 0, then \(\sqrt{b^{2} - 4ac}\) will not be real because \((\sqrt{b^{2} - 4ac})^{2}\) = b\(^{2}\) - 4ac < 0 and square of a real number always positive.
Thus, the roots of the equation ax\(^{2}\) + bx + c = 0 are not real if b\(^{2}\) - 4ac < 0.
As the value of b\(^{2}\) - 4ac determines the nature of roots (solution), b\(^{2}\) - 4ac is called the discriminant of the quadratic equation.
Definition of discriminant: For the quadratic equation ax\(^{2}\) + bx + c =0, a ≠ 0; the expression b\(^{2}\) - 4ac is called discriminant and is, in general, denoted by the letter ‘D’.
Thus, discriminant D = b\(^{2}\) - 4ac
Note:
Discriminant of ax\(^{2}\) + bx + c = 0 |
Nature of roots of ax\(^{2}\) + bx + c = 0 |
Value of the roots of ax\(^{2}\) + bx + c = 0 |
b\(^{2}\) - 4ac = 0 |
Real and equal |
- \(\frac{b}{2a}\), -\(\frac{b}{2a}\) |
b\(^{2}\) - 4ac > 0 |
Real and unequal |
\(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) |
b\(^{2}\) - 4ac < 0 |
Not real |
No real value |
When a quadratic equation has two real and equal roots we say that the equation has only one real solution.
Solved examples to examine the nature of roots of a quadratic equation:
1. Prove that the equation 3x\(^{2}\) + 4x + 6 = 0 has no real roots.
Solution:
Here, a = 3, b = 4, c = 6.
So, the discriminant = b\(^{2}\) - 4ac
= 4\(^{2}\) - 4 ∙ 3 ∙ 6 = 36 - 72 = -56 < 0.
Therefore, the roots of the given equation are not real.
2. Find the value of ‘p’, if the roots of the following
quadratic equation are equal (p - 3)x\(^{2}\) + 6x + 9 = 0.
Solution:
For the equation (p - 3)x\(^{2}\) + 6x + 9 = 0;
a = p - 3, b = 6 and c = 9.
Since, the roots are equal
Therefore, b\(^{2}\) - 4ac = 0
⟹ (6)\(^{2}\) - 4(p - 3) × 9 = 0
⟹ 36 - 36p + 108 = 0
⟹ 144 - 36p = 0
⟹ -36p = - 144
⟹ p = \(\frac{-144}{-36}\)
⟹ p = 4
Therefore, the value of p = 4.
3. Without solving the equation 6x\(^{2}\) - 7x + 2 = 0, discuss the nature of its roots.
Solution:
Comparing 6x\(^{2}\) - 7x + 2 = 0 with ax\(^{2}\) + bx + c = 0 we have a = 6, b = -7, c = 2.
Therefore, discriminant = b\(^{2}\) – 4ac = (-7)\(^{2}\) - 4 ∙ 6 ∙ 2 = 49 - 48 = 1 > 0.
Therefore, the roots (solution) are real and unequal.
Note: Let a, b and c be rational numbers in the equation ax\(^{2}\) + bx + c = 0 and its discriminant b\(^{2}\) - 4ac > 0.
If b\(^{2}\) - 4ac is a perfect square of a rational number then \(\sqrt{b^{2} - 4ac}\) will be a rational number. So, the solutions x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) will be rational numbers. But if b\(^{2}\) – 4ac is not a perfect square then \(\sqrt{b^{2} - 4ac}\) will be an irrational numberand as a result the solutions x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) will be irrational numbers. In the above example we found that the discriminant b\(^{2}\) – 4ac = 1 > 0 and 1 is a perfect square (1)\(^{2}\). Also 6, -7 and 2 are rational numbers. So, the roots of 6x\(^{2}\) – 7x + 2 = 0 are rational and unequal numbers.
Quadratic Equation
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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