# If 1 is a root of the quadratic equation \[{\text{3}}{{\text{x}}^{\text{2}}}{\text{ + ax - 2 = 0}}\] and the quadratic equation \[{\text{a}}\left( {{{\text{x}}^{\text{2}}}{\text{ + 6x}}} \right){\text{ - b = 0}}\] has equal roots, then what will be the value of b ?

Last updated date: 24th Mar 2023

•

Total views: 306.9k

•

Views today: 7.84k

Answer

Verified

306.9k+ views

**Hint:**Let us first, find the value of a and then put the value of a in the condition of equal roots to get the value of b.

**Complete step-by-step answer:**

Let, \[{\text{3}}{{\text{x}}^{\text{2}}}{\text{ + ax - 2 = 0}}\] (1)

And, \[{\text{a}}{{\text{x}}^{\text{2}}}{\text{ + 6ax - b = 0}}\] (2)

As it is given in the question that 1 is the root of equation 1.

And we know the condition of the roots of the equation that if p will be the root of any polynomial equation then p must satisfy that polynomial equation.

So, putting the value of x equal to 1 in equation 1. We get,

\[{\text{3}}{\left( 1 \right)^{\text{2}}}{\text{ + a}}\left( 1 \right){\text{ - 2 = 0}}\]

3 + a – 2 = 1 + a = 0

So, a = -1

Now, we had to find the value of b using equation 2.

So, as we know that for any quadratic equation \[p{x^{\text{2}}}{\text{ + qx + r = 0}}\]. If it has equal roots, then its determinant must be equal to zero.

And determinant of this quadratic equation will be \[{{\text{q}}^{\text{2}}}{\text{ - 4pr}}\]

As equation 2 has equal roots.

So, determinant of equation 2 will also be zero.

So, \[{\left( {{\text{6a}}} \right)^{\text{2}}}{\text{ - 4a}}\left( {{\text{ - b}}} \right){\text{ = 0}}\] (3)

Now putting the value of a in equation 3. We get,

\[{\left( {{\text{6}}\left( {{\text{ - 1}}} \right)} \right)^{\text{2}}}{\text{ - 4}}\left( {{\text{ - 1}}} \right)\left( {{\text{ - b}}} \right){\text{ = 0}}\]

\[{\left( {{\text{ - 6}}} \right)^{\text{2}}}{\text{ - 4b = 0}}\]

36 = 4b

Therefore, b = 9

**Hence, the value of b will be equal to 9.**

**Note:**Whenever we came up with this type of problem first, we had to find the value of a by putting the value of the given root of the equation to that equation. And after that we had to put the determinant of another given equation equal to zero because if roots of any quadratic equation are equal then its determinant must be equal to zero. This will be the easiest and efficient way to find the value of b.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE