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Maths > Quadratic Equations > Nature of Roots
Quadratic Equations

Nature of Roots

Can every quadratic equation be solved? Does a quadratic equation always have more than one solutions? Are there any equations that don’t have any real solution? The value of the variable for which the equation gets satisfied is called the solution or the root of the equation. The Nature of Roots of a quadratic equation is very interesting. Let us find out how!

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Roots of a Quadratic Equation

The number of roots of a polynomial equation is equal to its degree. Hence, a quadratic equation has 2 roots. Let α and β be the roots of the general form of the quadratic equation :ax2 + bx + c = 0. We can write:

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.

Nature of Roots

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.


quadratic equation cheat sheet

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: b– 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: b– 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  Real and unequal
b2 – 4ac = 0  Real and equal
b– 4ac < 0 Unequal and Imaginary
b2 – 4ac > 0 (is a perfect square)  Real, rational and unequal
b2 – 4ac > 0 (is not a perfect square)  Real, irrational and unequal
b2 – 4ac > 0 (is aperfect square and a or b is irrational)  Irrational

Let us put this to practice

Example 1: Discuss the nature of the roots of the quadratic equation 2x– 8x + 3 = 0.
Solution: Here the coefficients are all rational. The discriminant D of the given equation is
D = b– 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 4x– 4x +  1 = 0?
Solution: The discriminant D of the given equation is
D = b– 4ac
= (-4)– (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 =  0 [Because the roots are equal]
Therefore, p2 – 4(2)(8) = 0  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.

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vee kwari

how do we state the nature of quadratics

Karthik
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Karthik

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)

Vikas
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Vikas

HI

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