Question

the ratio of the sum and poduct of roots of the equation 7x square -12x +18 =0 is ? a)7:12 b)7:18 c)3:2 d)2:3

Answer 1

GIVEN : -

• Quadratic equation is 7x - 12x + 18 = 0

TO FIND : -

• Ratio of Sum of roots and Product of roots

SOLUTION :-

The general form of a quadratic equation is ax² + bx + c = 0 and let us assume that these roots are h , k

Then relation between h , k , a , b , c is given by

$\large\tt{(h+k)\:=\:\frac{-b}{a}}$

$\large\tt{(hk)\:=\:\frac{c}{a}}$

Given equation is 7x²-12x + 18 = 0

Let the roots of the given equation be λ , β

By comparing we get ,

• a = 7
• b = -12
• c = 18

$\large\tt{\rightarrow{(\lambda+\beta)\:=\:\frac{-(-12)}{7}\:=\:\frac{12}{7}}}$

$\large\tt{\rightarrow{(\lambda.\beta)\:=\:\frac{18}{7}}}$

We got the sum of the roots as 12/7 and product of roots is 18/7 .

$\large\tt{Ratio\:=\:\frac{12}{7}\::\:\frac{18}{7}}$

$\large\tt{\rightarrow{Ratio\:=\:12:18\:=\:2:3}}$

∴ The Ratio between the Sum of roots and product of roots of given quadratic equation is 2:3

Hence , Option(d) is correct

$\large\tt{\underline{\underline{\green{Additional\:Information:-}}}}$

❃ If in a quadratic equation ax² + bx + c = 0 , Discriminant i.e, b² - 4ac = 0 then roots of this equation are real and equal

• If b² - 4ac > 0 , Roots are real and distinct
• If b² - 4ac < 0 , Roots are not real

Answer 2

Question : -

The ratio of the sum and product of the roots of the quadratic equation 7x² - 12x + 18 = 0 is ?

• a ) 7 : 12

• b ) 7 : 18

• c ) 3 : 2

• d ) 2 : 3

Answer : -

Given : -

Quadratic equation : 7x² - 12x + 18 = 0

Required to find : -

• Ratio of the sum and product of the roots ?

Conditions used : -

Here, conditions refer to the relationship between the zeroes and the coefficients of the quadratic equation .

$\boxed{\tt{ \alpha + \beta = \dfrac{- coefficient \ of \ x^2 }{Coefficient \ of \ x }}}$

$\boxed{\tt{ \alpha \beta = \dfrac{ constant \ term }{ Coefficient \ of \ x^2 } }}$

Solution : -

Quadratic equation : 7x² - 12x + 18 = 0

We need to find the ratio of the sum and product of the roots ?

So,

Let find the roots of the equation .

The standard form of the Quadratic equation is ax² + bx + c = 0

Now,

Let's compare this standard form of the Quadratic equation with the given polynomial .

Hence,

• a = 7

• b = - 12

• c = 18

Let's find the value of the sum of the roots and the product of the roots .

Here,

Let's consider that alpha ( α ) , beta ( β ) are the roots of the quadratic equation .

We know that ;

The relationship between the sum of the zeroes and the coefficients is ;

$\boxed{\tt{ \alpha + \beta = \dfrac{- coefficient \ of \ x^2 }{Coefficient \ of \ x }}}$

So,

This implies ;

➪ α + β = - b/a

➪ α + β = - ( - 12 )/7

➪ α + β = 12/7

Hence,

• Value of sum of the roots = 12/7

Similarly,

The relationship between the product of the zeroes and the coefficients is ;

$\boxed{\tt{ \alpha \beta = \dfrac{ constant \ term }{ Coefficient \ of \ x^2 } }}$

This implies ;

➪ α.β = c/a

➪ α.β = 18/7

Hence,

• Value of product of the roots = 18/7

Now,

Let's find the ratio of the sum of the roots is to product of the roots

This implies ;

➪ α + β : α.β

➪ 12/7 : 18/7

However,

we know that ;

a : b can also be written as a/b

So,

➪ α + β : α.β

➪ 12/7 ÷ 18/7

➪ 12/7 x 7/18

➪ 12/18

➪ 12 : 18

By reducing using 6 table we get ;

➪ 2 : 3

Therefore,

The ratio of the sum and the product of the roots is 2 : 3

Hence,

Option - d is correct ✓