Browse more Topics Under Quadratic Equations
- Introduction to Quadratic Equations
- Solving Quadratic Equations
- Application of Quadratic Equations
- Nature of Roots
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Roots of a Quadratic Equation
α = (-b-√b2-4ac)/2a         and           β = (-b+√b2-4ac)/2a
Here a, b, and c are real and rational. Hence, the nature of the roots α and β of equation ax2 + bx + c = 0 depends on the quantity or expression (b2 – 4ac) under the square root sign. We say this because the root of a negative number can’t be any real number. Say x2 = -1 is a quadratic equation. There is no real number whose square is negative. Therefore for this equation, there are no real number solutions.
Hence, the expression (b2 – 4ac) is called the discriminant of the quadratic equation ax2 + bx + c = 0. Its value determines the nature of roots as we shall see. Depending on the values of the discriminant, we shall see some cases about the nature of roots of different quadratic equations.
Nature Of Roots
Let us recall the general solution, α = (-b-√b2-4ac)/2a and β = (-b+√b2-4ac)/2a
- Case I: b2 – 4ac > 0
When a, b, and c are real numbers, a ≠0 and the discriminant is positive, then the roots α and β of the quadratic equation ax2 +bx+ c = 0 are real and unequal.
- Case II: b2– 4ac = 0
When a, b, and c are real numbers, a ≠0 and the discriminant is zero, then the roots α and β of the quadratic equation ax2+ bx + c = 0 are real and equal.
- Case III: b2– 4ac < 0
When a, b, and c are real numbers, a ≠0 and the discriminant is negative, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are unequal and not real. In this case, we say that the roots are imaginary.
- Case IV: b2 – 4ac > 0 and perfect square
When a, b, and c are real numbers, a ≠0 and the discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real, rational and unequal.
- Case V: b2Â – 4ac > 0 and not perfect square
When a, b, and c are real numbers, a ≠0 and the discriminant is positive but not a perfect square then the roots of the quadratic equation ax2 + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.
- Case VI: b2Â – 4ac > 0 is perfect square and a or b is irrational
When a, b, and c are real numbers, a ≠0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax2 + bx + c = 0 are irrational.
Let us just summarize all the above cases in this table below:
| b2 – 4ac > 0 | Â Real and unequal |
| b2 – 4ac = 0 | Â Real and equal |
| b2Â – 4ac < 0 | Unequal and Imaginary |
| b2 – 4ac > 0 (is a perfect square) | Â Real, rational and unequal |
| b2 – 4ac > 0 (is not a perfect square) | Â Real, irrational and unequal |
| b2 – 4ac > 0 (is aperfect square and a or b is irrational) | Â Irrational |
Let us put this to practice
Example 1: Discuss the nature of the roots of the quadratic equation 2x2Â – 8x + 3 = 0.
Solution: Here the coefficients are all rational. The discriminant D of the given equation is
D = b2Â – 4ac = (-8)2 – 4 x 2 x 3
= 64 – 24
= 40 > 0
Clearly, the discriminant of the given quadratic equation is positive but not a perfect square. Therefore, the roots of the given quadratic equation are real, irrational and unequal.
Example 2: Without solving, examine the nature of roots of the equation 4x2Â – 4x +Â 1 = 0?
Solution: The discriminant D of the given equation is
D = b2Â – 4ac
= (-4)2Â – (4 x 4 x 1)
= 16-16=0
Clearly, the discriminant of the given quadratic equation is zero. Therefore, the roots are real and equal. Hence, here we have understood the nature of roots very clearly.
What are three different methods to solve Quadratic Equations?
More Solved Examples For You
Example 3: Determine the value(s) of p for which the quadratic equation 2x2 + px + 8 = 0 has equal roots:
A) p = ±64  B) p = ±8    C) p = ±4     D) p = ±16
Soution: B) The Discriminant of the given equation =Â 0 [Because the roots are equal]
Therefore, p2 – 4(2)(8) = 0Â or p2 = 64
Thus, p = ±8
Solved Questions for You
Question 1: What are the nature of roots?
Answer: The nature of roots simply means the category in which the roots are falling upon. The roots may be imaginary, real, unequal or equal. If the discriminate is negative, the roots will be imaginary.
Question 2: What does a negative discriminant mean?
Answer: A positive discriminant denotes that the quadratic has two different real number solutions. A discriminant of zero denotes that the quadratic consists of a repeated real number solution. A negative discriminant denotes that neither of the solutions is real numbers.
Question 3: What is a negative quadratic?
Answer: A quadratic expression that always takes positive values is referred to as positive definite, while one that always takes negative values is referred to as negative definite. Moreover, quadratics of either type do not ever take the value 0, thus their discriminant is negative.
Question 4: Why is the discriminant important?
Answer: The quadratic equation discriminant is significant since it tells us the number and kind of solutions. This information is useful as it serves as a double check when we solve quadratic equations by any of the four techniques. The four techniques are factoring, completing the square, using square roots, and using the quadratic formula.
how do we state the nature of quadratics
A polynomial can have real numbers as zero(ie rational and irrational)to decide it’s nature we can use the relationship between the coefficients and zero (by comparing sum and product of zeroes)