What is the nature of the roots of the quadratic equation if the value of its discriminant is zero

Answer

What is the nature of the roots of the quadratic equation if the value of its discriminant is zero
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Hint: The nature of the roots of a quadratic equation can be determined from the value of the discriminant. The roots of an equation are the points at which the curve meets the x-axis. So, we can find the nature of the root and thus predict the graph.

Complete step by step answer:

We know that the nature of the roots of a quadratic equation can be found from its discriminant. If the discriminant is greater than zero, the equation will have two real and distinct roots. If the discriminant is zero, the equation will have a real root. If the discriminant is less than zero, the equation will have no real roots, it will have 2 complex roots.
Graphically, the roots of an equation can be defined as the points where the curve of the equation meets the x-axis.
 So, if an equation has 2 roots, then the curve meets the x-axis at two points. Its graph is given by,

What is the nature of the roots of the quadratic equation if the value of its discriminant is zero


If the equation has no real then the curve does not meet the x-axis and its graph is given by,

What is the nature of the roots of the quadratic equation if the value of its discriminant is zero


 And if the equation has only 1 root, the graph meets the x-axis at only one point. Its graph is given by,

What is the nature of the roots of the quadratic equation if the value of its discriminant is zero


We are given that the quadratic equation has discriminant zero. So, the equation has only one root. So, the graph of the equation will touch the x-axis only once.
Therefore, the correct answer is option B.
Note: Discriminant of quadratic equation of the form $a{x^2} + bx + c = 0$ is given by $D = {b^2} - 4ac$.
If $D > 0$, the equation has 2 real and distinct roots. They are given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
If $D = 0$, the equation has a real root, which is given by, $x = \dfrac{{ - b}}{{2a}}$
If $D < 0$, the equation has complex roots.

The quadratic formula

The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.

The quadratic formula

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

Solving Quadratic Equations

General Properties of Quadratic Equation

Methods of Solving Quadratic Equations

Roots of a Quadratic Equation

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

9th Grade Math

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What is the nature of the roots of the quadratic equation whose discriminant is zero?

Clearly, the discriminant of the given quadratic equation is zero. Therefore, the roots are real and equal.

What is the nature of roots of the quadratic equation if the value of its discriminant is negative or less than zero?

The roots of a quadratic equation are imaginary and distinct if the discriminant of a quadratic equation is negative.

What is the nature of the roots of a quadratic equation?

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.

When the value of the discriminant is 0 the roots are?

When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two real roots.