Question:
The number of real values of \(m\) so that the equation
\[
x^2+(2m+1)x+m=0
\]
has equal roots is:
The number of real values of \(m\) so that the equation
\[
x^2+(2m+1)x+m=0
\]
has equal roots is:
Show Hint
A quadratic equation has equal roots only when its discriminant satisfies
\[
b^2-4ac=0.
\]
Always simplify the discriminant carefully before checking the nature of solutions.
Updated On: Jun 24, 2026
- \(1\)
- \(0\)
- \(2\)
- \(3\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Condition for equal roots.
For a quadratic equation \[ ax^2+bx+c=0, \] equal roots occur when the discriminant is zero: \[ D=b^2-4ac=0 \] Here, \[ a=1,\quad b=2m+1,\quad c=m \] Therefore, \[ (2m+1)^2-4(1)(m)=0 \]
Step 2: Simplify the equation.
Expanding, \[ 4m^2+4m+1-4m=0 \] \[ 4m^2+1=0 \] \[ 4m^2=-1 \] \[ m^2=-\frac14 \]
Step 3: Check for real values of \(m\).
Since \[ m^2=-\frac14 \] has no real solution, there is no real value of \(m\).
Hence, the number of real values of \(m\) is \[ 0 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{0} \]
For a quadratic equation \[ ax^2+bx+c=0, \] equal roots occur when the discriminant is zero: \[ D=b^2-4ac=0 \] Here, \[ a=1,\quad b=2m+1,\quad c=m \] Therefore, \[ (2m+1)^2-4(1)(m)=0 \]
Step 2: Simplify the equation.
Expanding, \[ 4m^2+4m+1-4m=0 \] \[ 4m^2+1=0 \] \[ 4m^2=-1 \] \[ m^2=-\frac14 \]
Step 3: Check for real values of \(m\).
Since \[ m^2=-\frac14 \] has no real solution, there is no real value of \(m\).
Hence, the number of real values of \(m\) is \[ 0 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{0} \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Quadratic Equations Questions
- If \( a \) is a common root of \( x^2 - 5x + \lambda = 0 \) and \( x^2 - 8x - 2\lambda = 0 \) (\( \lambda \neq 0 \)) and \( \beta, \gamma \) are the other roots of them, then \( a + \beta + \gamma + \lambda = \):
- If $\alpha$ and $\beta$ are two double roots of the equation: $$ x^2 + 3(a + 3)x - 9a = 0 $$ for different values of $a$ (where $\alpha > \beta$), then the minimum value of the equation: $$ x^2 + \alpha x - \beta = 0 $$ is:
- If \( 2x^2 + 3x - 2 = 0 \) and \( 3x^2 + \alpha x - 2 = 0 \) have one common root, then the sum of all possible values of \( \alpha \) is:
If $ \frac{k}{kx + 3} + \frac{3}{3x-k}= \frac{12x + 5}{(kx + 3)(3x - k)} $, then both the roots of the equation $ kx^2 - 7x + 3 = 0 $ are:
- If \( c \) and \( d \) are the roots of \( x^2 + ax + b = 0 \), then a root of the equation} \[ x^2 + (4c + a)x + (b + 2ac + 4c^2) = 0 \] is:
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions