Solving for m with Equal Roots
The problem requires finding the value(s) of m for the quadratic equation $(4 + m)x^2 + (m + 1)x + 1 = 0$ when its roots are equal.
Condition for Equal Roots
A quadratic equation $ax^2 + bx + c = 0$ has equal roots if its discriminant, $D$, is equal to zero. The discriminant is calculated as:
$D = b^2 - 4ac$
For the given equation, we identify the coefficients:
- $a = 4 + m$
- $b = m + 1$
- $c = 1$
Applying the Discriminant Condition
Setting the discriminant to zero:
$D = (m + 1)^2 - 4(4 + m)(1) = 0$
Calculating Values of m
Now, we solve the equation for m:
- Expand the terms: $(m^2 + 2m + 1) - (16 + 4m) = 0$
- Simplify the equation: $m^2 + 2m + 1 - 16 - 4m = 0$
- Combine like terms: $m^2 - 2m - 15 = 0$
- Factor the quadratic equation for m: $(m - 5)(m + 3) = 0$
- Determine the possible values for m: $m - 5 = 0$ or $m + 3 = 0$
- Therefore, the values are: $m = 5$ or $m = -3$.
Conclusion
The values of m for which the equation $(4 + m)x^2 + (m + 1)x + 1 = 0$ has equal roots are $5$ and $-3$.
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Important Questions from Quadratic Equation
If 2x 2+ 5x + 1 = 0, then one of the values of \(x - \frac{1}{{2x}}\) is:
If \(a-\frac{12}{a}=1\) , where a > 0, then the value of \(a^2+\frac{16}{a^2}\) is:
If x 2 – 3x + 1 = 0, then the value of \(\frac{(x^4+\frac{1}{x^2})}{(x^2+5x+1)}\) is:
If \(\sqrt{x}{}-{1\over\sqrt{x}}=\sqrt5\) , \(x \ne 0\) , then what is the value of \((x^4+{1\over{x^2}})\over(x^2+1) \) ?
If x 2 + \(\frac{1}{x^2}\) = 18, x > 0, then find the value of x 3 + \(\frac{1}{x^3}\) .
