Discriminant
Table of Contents
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Discriminant in Math
Discriminant in math is a function of the coefficients of the polynomial.
Discriminant Symbol
The symbol of discriminant is either \(\mathbf{D}\) or \(\mathbf{\Delta}\)
For example,
- the discriminant of a quadratic equation \(ax^2+bx+c=0\) is in terms of \(a,b,\) \( \text{ and }\) \(c\).
\[\Delta \text{ OR } D=b^2-4ac\] - the discriminant of a cubic equation \(ax^3+bx^2+cx+d=0\) is in terms of \(a,b,c\) \( \text{ and }\) \(d\).
\[\Delta\!\!=\!b^{2} c^{2}\!\!-\!4 a c^{3}\!\!-\!4 b^{3} d\!\!-\!27 a^{2} d^{2}\!\!+\!\!18 a b c d\]
We usually study about the discriminant of a quadratic equation.
Discriminant Formula
The discriminant, \(\mathbf{D}\) or \(\mathbf{\Delta}\), of a quadratic equation \(ax^2+bx+c=0\) is:
\(\mathbf{D}\) or \(\mathbf{\Delta=b^{2}-4 a c}\) |
Do you recall using \(b^2-4ac\) earlier?
Yes, it is a part of the quadratic formula:
\[x=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\]
Here, the expression that is inside the square root is called the discriminant of the quadratic equation.
The quadratic formula in terms of the discriminant is:
\[x=\dfrac{-b \pm \sqrt{D}}{2 a}\]
The discriminant can be either positive or negative or zero.
Discriminant Examples
Here is a discriminant example.
Example
Find the discriminant of \(2x^2+3x+3=0\)
Solution
Compare the given expression with \(ax^2+bx+c=0\),
\[a=2\\b=3\\c=3\]
The discriminant of the given equation is:
\[ \begin{aligned} D \text{OR}\Delta &= b^2-4ac\\[0.2cm] &= 3^2 -4(2)(3)\\[0.2cm] &=9-24\\[0.2cm] &=-15 \end{aligned}\]
Thus, the discriminant of the given equation is:
\(\mathbf{D}\) or \(\mathbf{\Delta=-15}\) |
Relation between Discriminant and Nature of the Roots
The roots of a quadratic equation \(ax^2+bx+c=0\) are the values of \(x\) that satisfy the equation.
They can be found using the quadratic formula:
\(x=\dfrac{-b \pm \sqrt{D}}{2 a}\) |
Though we cannot find the roots by just using the discriminant, we can determine the nature of the roots as follows:
- If \(D>0\), the quadratic equation has two different real roots:
\[ \dfrac{-b \pm \sqrt{ \text{Positive number}}}{2 a}\]
gives two roots
- If \(D=0\), the quadratic equation has only one real root:
\[ \dfrac{-b \pm \sqrt{0}}{2 a}= \dfrac{-b }{2 a}\]
is the only root
- If \(D<0\), the quadratic equation has no real roots. i.e., it has two complex roots:
\[ \dfrac{-b \pm \sqrt{ \text{Negative number}}}{2 a}\]
gives two complex roots.
This is because the square root of a negative number leads to an imaginary number. i.e., \(\sqrt{-1} = i\)
A root is nothing but the x-coordinate of the x-intercept.
The graph of a quadratic equation in each of these 3 cases can be as follows.
Discriminant Calculator (with Graph)
Here is the "Discriminant Calculator".
You can move the sliders to change the coefficients of a quadratic expression.
Then it shows the corresponding discriminant.and the graph.
It also shows the number of roots depending on the discriminant.
- The discriminant of a quadratic equation \(\mathbf{ax^2+bx+c=0}\) is \(D \text{ OR } \Delta = b^2-4ac\)
- A quadratic equation has:
(i) two unequal real roots when \(D>0\)
(ii) only one real root when \(D=0\)
(iii) no real roots or two complex roots when \(D<0\)
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Solved Examples
Here are some discriminant examples and their solutions.
Example 1 |
Find the discriminant of \(\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0\)
Solution:
Comparing the given equation with \(ax^2+bx+c=0\),
\[ \begin{aligned} a&= \sqrt{3}\\[0.2cm]b &=10\\[0.2cm]c&=-8\sqrt{3} \end{aligned} \]
The discriminant of the given equation is,
\[ \begin{aligned} D \text{ OR }\Delta &= b^2-4ac\\[0.2cm] &= (10)^2 -4( \sqrt{3})(-8\sqrt{3})\\[0.2cm] &=100+96\\[0.2cm]&=196 \end{aligned}\]
\(\mathbf{D}\) or \(\mathbf{\Delta=196}\) |
Example 2 |
Determine whether the following equation has two real roots, one real root or no real roots.
\[3x^2 - 5x -7=0\]
Solution:
Comparing the given equation with \(ax^2+bx+c=0\),
\[ \begin{aligned} a&=3\\[0.2cm]b &=-5\\[0.2cm]c&=-7\end{aligned} \]
The discriminant of the given equation is,
\[ \begin{aligned} D \text{ OR }\Delta &= b^2-4ac\\[0.2cm] &= (-5)^2 -4( 3)(-7)\\[0.2cm] &=25+84\\[0.2cm]&= \sqrt{109}\end{aligned}\]
Since the discriminant is positive here,
The number of real roots \(= 2\) |
Let us check it using the graph.
It has two x-intercepts which means that the number of real roots is \(2\).
Example 3 |
Find the discriminant of the following equation:
\[9 z^{2}-6 b^{2} z-\left(a^{4}-b^{4}\right)=0\]
Solution:
Comparing the given equation with \(Ax^2+Bx+C=0\),
\[ \begin{aligned} A&=9\\[0.2cm]B &=-6b^2\\[0.2cm]C&=-(a^4-b^{4})\end{aligned} \]
The discriminant of the given equation is,
\[ \begin{aligned} D \text{ OR }\Delta &= b^2-4ac\\[0.2cm] &= (-6b^2)^2 -4( 9)(-(a^4-b^{4}))\\[0.2cm] &=36b^4 +36a^4-36b^4\\[0.2cm]&= 36a^4\end{aligned}\]
\(\mathbf{D}\) or \(\mathbf{\Delta=36a^4}\) |
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Practice Questions
Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
1. | Identify the discriminant of \(\mathbf{x^{2}+5 x-\left(p^{2}+p-6\right)=0}\) from the following: | |
\(\begin{aligned}\mathbf{a.D\!=\!(2 p\!+\!1)}\end{aligned}\) | \(\begin{aligned}\mathbf{b.D\!=\!(2 p\!+\!1)^{2}}\end{aligned}\) | |
\(\begin{aligned}\mathbf{c.D\!=\!4 p^{2}}\end{aligned}\) | \(\begin{aligned}\mathbf{d.D\!=\!(2 p\!-\!7)^{2}}\end{aligned}\) | |
2. | Find the positive value of \(k\) such that the equation \(\mathbf{k x^{2}-24 x+2=0}\) has only one real solution. |
Maths Olympiad Sample Papers
IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.
You can download the FREE grade-wise sample papers from below:
- IMO Sample Paper Class 1
- IMO Sample Paper Class 2
- IMO Sample Paper Class 3
- IMO Sample Paper Class 4
- IMO Sample Paper Class 5
- IMO Sample Paper Class 6
- IMO Sample Paper Class 7
- IMO Sample Paper Class 8
- IMO Sample Paper Class 9
- IMO Sample Paper Class 10
To know more about the Maths Olympiad you can click here
Frequently Asked Questions (FAQs)
1. What is the discriminant formula?
The discriminant, \(\mathbf{D}\) or \(\mathbf{\Delta}\), of a quadratic equation \(ax^2+bx+c=0\) is:
\(\mathbf{D}\) or \(\mathbf{\Delta=b^{2}-4 a c}\) |
2. What is a discriminant example?
Example
Find the discriminant of \(\sqrt{3} x^{2}+11 x+10 \sqrt{3}=0\)
Solution
Compare the given expression with \(ax^2+bx+c=0\),
\[a=\sqrt{3}\\[0.2cm]b=11\\[0.2cm]c=10\sqrt{3}\]
The discriminant of the given equation is:
\[ \begin{aligned} D \text{ OR }\Delta &= b^2-4ac\\[0.2cm] &= (11)^2 -4(\sqrt{3})(10\sqrt{3})\\[0.2cm] &=121-120\\[0.2cm] &=1 \end{aligned}\]
Thus, the discriminant of the given equation is:
\(\mathbf{D}\) or \(\mathbf{\Delta=1}\) |
3. How do you use the discriminant formula?
We can use the discriminant to find the nature of the roots.
A quadratic equation has:
(i) two unequal real roots when \(D>0\)
(ii) only one real root when \(D=0\)
(iii) no real roots or two complex roots when \(D<0\)