In equation {x^3} + 3Hx + G = 0 , if G and H are real and {G^2} + 4{H^3}

English

Gujarati

Hindi

4-2.Quadratic Equations and Inequations

hard

In the equation ${x^3} + 3Hx + G = 0$, if $G$ and $H$ are real and ${G^2} + 4{H^3} > 0,$ then the roots are

All real and equal

All real and distinct

One real and two imaginary

All real and two equal

Solution

Let $\alpha ,\beta $ be the roots of given cubic equation.

We know that $\alpha = {\left( {\frac{{ – G + \sqrt {{G^2} + 4{H^3}} }}{2}} \right)^{1/3}}$ and $\beta = {\left( {\frac{{ – G – \sqrt {{G^2} + 4{H^3}} }}{2}} \right)^{1/3}}$, since ${G^2} + 4{H^3}> 0,$

therefore the cubic equation has got one real and two imaginary roots.

Standard 11

Mathematics

If the sum of two of the roots of ${x^3} + p{x^2} + qx + r = 0$ is zero, then $pq =$

medium

$\alpha$, $\beta$ ,$\gamma$ are roots of equatiuon $x^3 -x -1 = 0$ then equation whose roots are $\frac{1}{{\beta + \gamma }},\frac{1}{{\gamma + \alpha }},\frac{1}{{\alpha + \beta }}$ is

normal

Let $t$ be real number such that $t^2=a t+b$ for some positive integers $a$ and $b$. Then, for any choice of positive integers $a$ and $b, t^3$ is never equal to

normal

(KVPY-2016)

The number of roots of the equation $\log ( – 2x)$ $ = 2\log (x + 1)$ are

easy

If $x$ is real , the maximum value of $\frac{{3{x^2} + 9x + 17}}{{3{x^2} + 9x + 7}}$ is