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. Show 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. Case 1: D = 0 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. Case 2: D > 0 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. Case 3: D < 0 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. Case 4: D > 0 and perfect square If D > 0 and a perfect square, then the roots of the quadratic equation are real, unequal and rational. Case 5: D > 0 and not a perfect square 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.
Bridge Course – Nature of Roots of Quadratic Equations Also Read Quadratic inequalities Solved ExamplesExample 1: 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 Solution: 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. Example 2: 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 Solution: 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. Example 3: The quadratic equation with real coefficients when one of its root is (3+2i) is Solution: 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 Example 4: Show that the equation 3x2+4x+6 = 0 has no real roots. Solution: 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.
Answer: the solutions to the equation are not only real, but also rational. Step-by-step explanation:
The quadratic formula was covered in the module Algebra review. This formula gives solutions to the general quadratic equation \(ax^2+bx+c=0\), when they exist, in terms of the coefficients \(a,b,c\). The solutions are \[ x = \dfrac{-b+\sqrt{b^2-4ac}}{2a}, \qquad x = \dfrac{-b-\sqrt{b^2-4ac}}{2a}, \]provided that \(b^2-4ac \geq 0\). The quantity \(b^2-4ac\) is called the discriminant of the quadratic, often denoted by \(\Delta\), and should be found first whenever the formula is being applied. It discriminates between the types of solutions of the equation:
Screencast of interactive 2
For what values of \(k\) does the equation \((4k+1)x^2-6kx+4=0\) have one real solution? A quadratic expression which always takes positive values is called positive definite, while one which always takes negative values is called negative definite. Quadratics of either type never take the value 0, and so their discriminant is negative. Furthermore, such a quadratic is positive definite if \(a > 0\), and negative definite if \(a < 0\). Detailed description of diagram
Show that the quadratic expression \(4x^2-8x+7\) always takes positive values for any value of \(x\). SolutionIn this case, \(a=4\), \(b=-8\) and \(c=7\). So \[ \Delta = (-8)^2 - 4\times 4 \times 7 = -48 < 0 \]and \(a=4 > 0\). Hence the quadratic is positive definite.
For what values of \(k\) does the equation \((4k+1)x^2-2(k+1)x+(1-2k)=0\) have one real solution? For what values of \(k\), if any, is the quadratic negative definite? Screencast of exercise 5 Next page - Content - Applications to maxima and minima
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What is the nature of roots of the quadratic equation if the value of its discriminant is positive?
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