Discriminant of a Quadratic Equation
A quadratic equation's number of solutions is determined using the discriminant formula. The discriminant is the name given to the expression found under the square root (radical) symbol in the quadratic formula. For the general quadratic, or conic, equationax2+bxy+cy2+dx+ey+f=0, a discriminant can be found that indicates whether the conic depicted is an ellipse, a hyperbola, or a parabola. Applications of determinants can be seen in elliptic curves, finite field extensions, quadratic forms, and other mathematical entities.
Here, students will learn what is discriminant in quadratic equation and how to find the discriminant of the equation.
Discriminant of a Quadratic Equation
Discriminant, it is a part of the quadratic formula:
\[x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\]
In the above equation, the expression that is inside the square root is called the discriminant of a quadratic equation.
The quadratic formula in terms of the discriminant is given below:
\[x=\frac{-b\pm \sqrt{D}}{2a}\]
Formula of Discriminant
A polynomial's discriminant is a function of its coefficients and is denoted by a capital "D" or the Delta symbol (). It demonstrates the existence of the roots of any quadratic equation with rational numbers a, b, and c. A quadratic equation will easily display the real roots or the number of x-intercepts. This formula determines whether the quadratic equation's roots are real or imaginary.
The formula of discriminant in the quadratic equation ax2+bx+c=0 is
\[\Delta =b^{2}-4ac\]
The number of roots of a quadratic equation can be calculated using the discriminant. A discriminant may be positive, negative, or zero in value. The nature of roots can be calculated by knowing the meaning of a determinant as follows:
If the value of discriminant = 0 i.e. b2-4ac=0 | The quadratic equation will have equal roots i.e. α = β = -b/2a |
If the value of discriminant < 0 i.e. b2-4ac<0 | The quadratic equation will have imaginary roots i.e α = (p + iq) and β = (p – iq). Where ‘iq’ is the imaginary part of a complex number |
If the value of discriminant (D) > 0 i.e. b2-4ac>0 | The quadratic equation will have real roots |
If the value of discriminant > 0 and D is a perfect square | The quadratic equation will have rational roots |
If the value of discriminant (D) > 0 and D is not a perfect square | The quadratic equation will have irrational roots i.e. =(p+q) and =(p-q) |
If the value of discriminant > 0, D is a perfect square, then a = 1 , b and c are integers | The quadratic equation will have integral roots |
Solved Examples:
1.What is the discriminant of the equation x2-2x+3=0? Also, figure out how many solutions this equation has.
Sol: Given, x2-2x+3=0
In the equation, the value of a = 1 , b = -2 and c = 3
The formula for discriminant is,
Δ = b2-4ac
=> Δ = (-2)2-4(1)(3)
=>Δ = 4 – 12
Δ = -8 < 0
The equation would have no real solutions since the determinant's value is negative.
2.Find the discriminant of the following equation:
9z2-6b2z-(a4-b4)=0
Sol:Comparing the given equation with Ax2+Bx+C=0
Given A=9, B=-6b2and C=-(a4-b4)
The discriminant of the given equation is,
D or =b2-4ac
=(-6b2)2-4(9)(-(a4-b4))
=36b4+36a4-36b4
=36a4
Hence, the discriminant value is 36a4
3.Determine whether the following equation has two real roots, one real root, or no real roots.
3x2-5x-7=0
Sol: Comparing the given equation with ax2+bx+c=0
Here a=3, b=-5 and c=-7
The discriminant of the given equation is, b2-4ac
=(-5)2-4(3)(-7)
=25+84
=109
Since the discriminant is positive here,
Hence, the number of real roots will be 2
Conclusion
The discriminant is the under-the-radical element of the quadratic formula. It is used to calculate the number of roots you will have. After going through the above article students should be able to know what is the formula of discriminant and what is the discriminant of quadratic equation.
From above we can conclude that there are three possible outcomes for the discriminant:
Discriminant equationb2-4ac
Ifb2-4ac<0-No real roots. Roots will have imaginary numbers. This means the quadratic is either always above the axis, or always below.
If b2-4ac=0- One real root. The graph touches the x-axis in one place.
If b2-4ac>0- Two real roots. The graph crosses the x-axis twice.
FAQs on Discriminant Formula
1. How do you tell if a discriminant is rational or irrational?
The solutions to the equation are not only real but also rational if the discriminant is a perfect square. The solutions to the equation are true but irrational if the discriminant is positive but not a perfect square.
2. What does it mean if the discriminant is less than 0?
A quadratic function with a discriminant less than zero has no real origin, and the parabola it represents does not intersect the x-axis.
3. What if the discriminant is negative?
If the discriminant is negative, the quadratic formula contains a negative number under the square root. The imaginary number i would be involved in the square root of a negative number. As a result, we will have two complex solutions.
