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. Show 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 = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).
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.
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.
Thus, discriminant D = b\(^{2}\) - 4ac
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:
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.
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.
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.
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.
We know that a quadratic equation is a second degree polynomial equation of the form ax2 + bx + c = 0, where a, b, c are real numbers, x is the unknown variable and a ≠ 0. For the equation ax2 + bx + c = 0, the discriminant is given by D = b2 – 4ac. It is also denoted by ∆. A quadratic equation has 2 roots. It will be real or imaginary. In this article we discuss the nature of roots depending upon coefficients and discriminant. If α and β are the values of x which satisfy the quadratic equation, α and β are called the roots of the quadratic equation. Roots are given by the equation (-b±√(b2-4ac))/2a. The nature of the roots depends on the discriminant. ## Nature of Roots depending upon DiscriminantAccording to the value of discriminant, we shall discuss the following cases about the nature of roots.
If the discriminant is equal to zero (b2 – 4ac = 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax2 + bx + c = 0, are real and equal. In this case, the roots are x = -b/2a. The graph of the equation touches the X axis at a single point.
If the discriminant is greater than zero (b2 – 4ac > 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax2 + bx + c = 0, are real and unequal. The graph of the equation touches the X-axis at two different points.
If the discriminant is less than zero (b2 – 4ac < 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax2 + bx + c = 0, are imaginary and unequal. The roots exist in conjugate pairs. The graph of the equation does not touch the X-axis.
If D > 0 and a perfect square, then the roots of the quadratic equation are real, unequal and rational.
If D > 0 and not a perfect square, then the roots of the quadratic equation are real, unequal and irrational. We can summarize all the above cases in the table below.
## Nature of Roots depending upon coefficientsDepending upon the nature of the coefficients of the quadratic equation, we can summarize the following. - If c = 0, then one of the roots of the quadratic equation is zero and the other is -b/a.
- If b = c = 0, then both the roots are zero.
- If a = c, then the roots are reciprocal to each other.
## Bridge Course – Nature of Roots of Quadratic Equations
Quadratic inequalities ## Solved Examples
The roots of the quadratic equation 3x2-10x+3 = 0 are a) real and equal b) imaginary c) real, unequal and rational d) none of these
Given equation 3x2-10x+3 = 0 Here discriminant, D = b2-4ac => (-10)2 – 4×3×3 = 100 – 36 = 64 D is positive and a perfect square. So the roots of the quadratic equation are real, unequal and rational. Hence option c is the answer.
Find the value of p if the equation 3x2-18x+p = 0 has real and equal roots. a) 27 b) 18 c) 9 d) none of these
Given 3x2-18x+p = 0 has real and equal roots. => b2-4ac = 0 =>(-18)2-4×3×p = 0 => 324 – 12p = 0 => p = 324/12 = 27 Hence option a is the answer.
The quadratic equation with real coefficients when one of its root is (3+2i) is
Given one root is 3+2i. Complex roots occur in conjugate pairs. So other root = 3-2i Sum of roots = 6 Product of roots = (3+2i)(3-2i) = 13 Required equation is x2-(Sum)x+Product = 0 => x2-6x+13 = 0
Show that the equation 3x2+4x+6 = 0 has no real roots.
Given equation 3x2+4x+6 = 0 Here a = 3, b = 4, c = 6 Discriminant D = b2-4ac => 42-4×3×6 = 16-72 = -56 Since D<0, the roots are imaginary. Hence the equation has no real roots. ## Video Lesson – Nature of Roots
The discriminant of a quadratic equation is given by D = b2 – 4ac.
If discriminant, D = 0, then the roots are real and equal.
If discriminant, D>0, then the roots are real and unequal.
If discriminant, D<0, then the roots are imaginary and unequal. |