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An algebraic statement of the second degree in x is called a quadratic equation. ax² + bx + c = 0 is the standard form of the quadratic equation, where a, b are the coefficients, x is the variable, and c is the constant term. The coefficient of x² is a non-zero term(a 0), which is the first criterion for an equation to be a quadratic equation. The x² term is written first, then the x term, and finally the constant term when expressing a quadratic equation in standard form. The numeric values of a, b, and c are usually expressed as integral values rather than fractions or decimals.

Table of Content

Discriminant Formula for a Quadratic Equation
Examples using Discriminant Formula
Importance of Discriminant Formula
Things to Remember
Sample Questions

Key Takeaways: Quadratic, Root, equation, positive, negative, solution, discriminant, polynomial

Discriminant Formula for a Quadractic Equation

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The discriminant is the b²-4ac component of the quadratic formula beneath the square root sign. The discriminant indicates whether there are one, two, or no solutions available.

The solutions can be found by the formula

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

x is the solution.

b is the coefficient of the equation where the variable is in the power of 1.

c is the coefficient of the equation where the variable is in the power of 2.

a is the coefficient of the equation where the variable is in the power of 0.

Example using Discriminant Formula

Question: Find the discriminant of the equation x² – 2x + 3 = 0. Determine the number of solutions to the equation.

Solution: x² – 2x + 3 = 0

In the above equation,

a = 1 , b = -2 , and c = 3

The formula for discriminant is b²-4ac,

Δ = b²-4ac

=> Δ = 2² – 4(1)(3)

=>Δ = 4 – 12

Δ = -8 < 0

Since the value of the determinant is negative, the equation has no real solutions.

The video below explains this:

Quadratic Equations Detailed Video Explanation:

Also Read:

Equation of a Line	Straight Lines	Limit Formula

Importance of Discriminants

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The number of roots of a quadratic equation can be found using the discriminant. A discriminant might be positive, negative, or 0 in value. The nature of roots can be established by knowing the value of a determinant as follows:

The quadratic equation has two actual and distinct solutions if the discriminant value is positive.
The quadratic equation has only one or two real and equal solutions if the discriminant value is zero.
The quadratic equation has no real solutions if the discriminant value is negative.

Things to Remember

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An algebraic statement of the second degree in x is called a quadratic equation. ax² + bx + c = 0 is the standard form of the quadratic equation, where a, b are the coefficients, x is the variable, and c is the constant term.
The formula for discriminant is b²-4ac.
If the determinant, Δ > 0, the equation will have 2 real solutions.
If the determinant, Δ = 0, the equation will have 1 real solution.
If the determinant, Δ < 0, the equation will have 2 conjugate complex solutions.

Also Read:

Integer as Exponents	Ordinate	Collinear Points
Midpoint Formula	Section Formula	Sphere Formula

Sample Questions

Ques. Find the discriminant of the quadratic equation 5x² + 3x + 2 = 0. Also, find out the nature of its roots. (3 marks)

Ans. The quadratic equation given is 5x² + 3x + 2 = 0.

Matching this with the quadratic equation formula ax² + bx + c = 0, we get the following values:

a = 5, b = 3, and c = 2.

Using the discriminant formula b²-4ac,

Δ = b²-4ac

= 3² - 4(5)(2)

= 9 - 40

= -31

Therefore, the discriminant is -31 which is a negative number and hence the given quadratic equation has two conjugate complex solutions.

Ques. Find the discriminant of the quadratic equation 3x² + 4x + 5 = 0. Also, find out the nature of its roots. (3 marks)

Ans. The quadratic equation given is 5x² + 3x + 2 = 0.

Matching this with the quadratic equation formula ax² + bx + c = 0, we get the following values:

a = 5, b = 3, and c = 2.

Using the discriminant formula b²-4ac,

Δ = b²-4ac

= 5² - 4(5)(2)

= 25 - 40

= -15

Therefore, the discriminant is -15 which is a negative number and hence the given quadratic equation has two conjugate complex solutions.

Ques. Find the discriminant of the quadratic equation x² + 4x - 5 = 0. Also, find out the nature of its roots. (3 marks)

Ans. The quadratic equation given is x² + 4x - 5 = 0.

Matching this with the quadratic equation formula ax2 + bx + c = 0, we get the following values:

a = 1, b = 4, and c = -5.

Using the discriminant formula b²-4ac,

Δ = b²-4ac

= 4² - 4(1)(-5)

= 16 + 20

= 36

Therefore, the discriminant is 36 which is a positive number and hence the given quadratic equation has two real solutions.

The equation has the following two solutions:

(-4 36)/2 which gives x = 1 or x = -5

Ques. Find the discriminant of the quadratic equation 2x² + 8x + 8 = 0. Also, find out the nature of its roots. (3 marks)

Ans. The quadratic equation given is 2x² + 8x + 8 = 0.

Matching this with the quadratic equation formula ax² + bx + c = 0, we get the following values:

a = 2, b = 8, and c = 8.

Using the discriminant formula b²-4ac

Δ = b²-4ac

= 8² - 4(2)(8)

= 0

Therefore, the discriminant is 0 which signifies that the given quadratic equation will have 1 real solution.

Ques. Find the discriminant of the quadratic equation 6x² + 10x - 1 = 0. Also, find out the nature of its roots. (3 marks)

Ans. The quadratic equation given is 6x² + 10x - 1 = 0.

Matching this with the quadratic equation formula ax² + bx + c = 0, we get the following values:

a = 6, b = 10, and c = -1.

Using the discriminant formula b²-4ac

Δ = b²-4ac

= 10² - 4(6)(-1)

= 100 + 24

= 124 > 0

Therefore, the discriminant is 124 which is a positive number and hence the given quadratic equation has two real solutions.

Ques. Find the discriminant of the quadratic equation x² - 4 = 0. Also, find out its roots. (3 marks)

Ans. The quadratic equation given is x² - 4 = 0.

Matching this with the quadratic equation formula ax² + bx + c = 0, we get the following values:

a = 1, b = 0, and c = -4.

Using the discriminant formula b²-4ac,

Δ = b²-4ac

= 0² - 4(1)(-4)

= 16

= 16 > 0

Therefore, the discriminant is 16 which is a positive number and hence the given quadratic equation has two real solutions.

The equation has the following two solutions:

(-0 16)/2 which gives x = 2 or x = -2

Ques. Find the discriminant of the quadratic equation 2x² - 8x + 3 = 0. Also, find out the nature of its roots. (3 marks)

Ans. The quadratic equation given is 2x² + 8x + 8 = 0.

Matching this with the quadratic equation formula ax² + bx + c = 0, we get the following values:

a = 2, b = -8, and c = 3.

Using the discriminant formula b²-4ac,

Δ = b²-4ac

= (-8)² - 4(2)(3)

= 40 > 0

Therefore, the discriminant is 40 which is a positive number and hence the given quadratic equation has two real solutions.

Discriminant Formula for a Quadratic Equation: Definition, Example and Importance

Discriminant Formula for a Quadractic Equation

Example using Discriminant Formula

Quadratic Equations Detailed Video Explanation:

Importance of Discriminants

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

4.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

5.
In the given figure, point D divides the side BC of $\Delta ABC$ in the ratio $1 : 2$. Find length AD.

6.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

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Discriminant Formula for a Quadratic Equation: Definition, Example and Importance

Discriminant Formula for a Quadractic Equation

Example using Discriminant Formula

Quadratic Equations Detailed Video Explanation:

Importance of Discriminants

Things to Remember

Sample Questions

CBSE X Related Questions

1.Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent. Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

2.In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

4.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

5.In the given figure, point D divides the side BC of $\Delta ABC$ in the ratio $1 : 2$. Find length AD.

6.If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

4.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

5.
In the given figure, point D divides the side BC of $\Delta ABC$ in the ratio $1 : 2$. Find length AD.

6.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals