The real number k for which the equation 2x^2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $f(x) = 2x^2 + 3x + k$.For a quadratic equation $ax^2 + bx + c = 0$ to have two distinct real roots,the discriminant $D = b^2 - 4ac$ must be g

Vedclass · Accepted Answer

does not exist

Vedclass · Answer

(D) Let $f(x) = 2x^2 + 3x + k$.For a quadratic equation $ax^2 + bx + c = 0$ to have two distinct real roots,the discriminant $D = b^2 - 4ac$ must be greater than $0$.Here,$a = 2, b = 3, c = k$.$D = 3^2 - 4(2)(k) = 9 - 8k$.For distinct real roots,$9 - 8k > 0 \Rightarrow k However,the roots of the quadratic equation $2x^2 + 3x + k = 0$ are given by $x = \frac{-3 \pm \sqrt{9 - 8k}}{4}$.For these roots to lie in the interval $[0, 1]$,we must have $0 \le \frac{-3 \pm \sqrt{9 - 8k}}{4} \le 1$.Since $x$ must be real,$9 - 8k \ge 0$. If we take the root $x = \frac{-3 + \sqrt{9 - 8k}}{4}$,for this to be $\ge 0$,we need $\sqrt{9 - 8k} \ge 3$,which implies $9 - 8k \ge 9$,so $k \le 0$.If $k \le 0$,then the other root $x = \frac{-3 - \sqrt{9 - 8k}}{4}$ will be $\le -3/4$,which is outside the interval $[0, 1]$.Thus,it is impossible for both roots to lie in $[0, 1]$ simultaneously.Therefore,such a real number $k$ does not exist.

The real number $k$ for which the equation $2x^2 + 3x + k = 0$ has two distinct real roots in $[0, 1]$:

Similar Questions

Sum of the solutions of the equation $[x^2] - 2x + 1 = 0$ is (where $[.]$ denotes the greatest integer function).

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x + 4 = 0$,then the equation whose roots are $\alpha^2$ and $\beta^2$ is

The number of solutions of $\frac{\log 5 + \log (x^2 + 1)}{\log (x - 2)} = 2$ is

If $\frac{m-a^{2}}{b^{2}+c^{2}}+\frac{m-b^{2}}{c^{2}+a^{2}}+\frac{m-c^{2}}{a^{2}+b^{2}}=3,$ then the value of $m$ is

The value of $x$ in the equation $\left(x+\frac{1}{x}\right)^{2}-\frac{3}{2}\left(x-\frac{1}{x}\right)=4$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module