Question

If alpha and beta are the roots of the equation 3x²+7x-2=what are the value of alpha)beta+beta/alpha

Answer 1

Answer:

alfa= -7+root73/6 ,beta= -7 -root 73/6

Step-by-step explanation:

3x^2+7x-2=0

comparing with ax^2 +bx+c=0

a=3 ,b= 7, c= -2

b^2 -4ac= 7^2 -( 4×3× -2)=49+24 =73

x = -b +- root ( b^2 -4ac)/2a

x= -7 +-root 73/ 6

alfa= -7 +-root 73/6

beta= -7 +- root 73/6

Answer 2

Appropriate Question :-

If $\alpha$ and $\beta$ are the roots of the equation 3x²+7x-2=0, what is the value of $\dfrac{\alpha}{\beta}+ \dfrac{\beta}{\alpha}$ ?

$\large\underline{\sf{Solution-}}$

Given that

$\rm \: \alpha, \beta \: are \: the \: roots \: of \: the \: equation \: {3x}^{2} + 7x - 2 = 0$

We know,

$\boxed{\red{\sf Sum\ of\ the\ roots=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

$\rm\implies \: \alpha + \beta = - \: \dfrac{7}{3}$

And

$\boxed{\red{\sf Product\ of\ the\ roots=\frac{Constant}{coefficient\ of\ x^{2}}}}$

$\rm\implies \: \alpha\beta = - \: \dfrac{2}{3}$

Now, Consider

$\rm \: \dfrac{ \alpha }{ \beta } + \dfrac{ \beta }{ \alpha }$

$\rm \: = \: \dfrac{ { \alpha }^{2} + { \beta }^{2} }{ \alpha \beta }$

$\rm \: = \: \dfrac{ ({ \alpha + \beta )}^{2} - 2 \alpha \beta }{ \alpha \beta }$

$\rm \: = \: \bigg[\bigg( - \dfrac{7}{3}\bigg)^{2} + 2 \times \dfrac{2}{3} \bigg] \div \bigg( - \dfrac{2}{3}\bigg)$

$\rm \: = \: \bigg[ \frac{49}{9} + \dfrac{4}{3} \bigg] \times \bigg( - \dfrac{3}{2}\bigg)$

$\rm \: = \: \bigg[ \frac{49 + 12}{9} \bigg] \times \bigg( - \dfrac{3}{2}\bigg)$

$\rm \: = \: \bigg[ \frac{61}{3} \bigg] \times \bigg( - \dfrac{1}{2}\bigg)$

$\rm \: = \: - \: \dfrac{61}{6}$

Hence,

$\rm\implies \:\boxed{ \rm{ \:\bf\: \dfrac{ \alpha }{ \beta } + \dfrac{ \beta }{ \alpha } = \: - \: \dfrac{61}{6} \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac