
If 1 is the root of the quadratic equation $3{x^2} + ax - 2 = 0$ and the quadratic equation $a({x^2} + 6x) - b = 0$ has equal roots, find the value of b.
Answer
458.4k+ views
Hint: According to given in the question we have to determine the value of b if 1 is the root of the quadratic equation $3{x^2} + ax - 2 = 0$ and the quadratic equation $a({x^2} + 6x) - b = 0$ has equal roots. So, first of all as mentioned in the question that 1 is the root of the given equation so it will satisfy the equation hence, we have to substitute this in the given expression which is $3{x^2} + ax - 2 = 0$ so that we can easily determine the value of a.
Now, as mentioned in the question that the quadratic expression $a({x^2} + 6x) - b = 0$ has two equal roots so we have to use the formula for the quadratic expression when the roots are equal to each other which is as mentioned below:
$ \Rightarrow {b^2} = 4ac..............(A)$
Hence, on substituting all the values in the formula (A) just above in the expression $a({x^2} + 6x) - b = 0$ we can determine the value of $b$and we have to also substitute the value of a as obtained with the help of the expression$3{x^2} + ax - 2 = 0$ which we have already obtained.
Complete answer:
Step 1: Since, as given in the question that 1 is the root of the quadratic equation $3{x^2} + ax - 2 = 0$so we have to put $x = 1$in the given expression$3{x^2} + ax - 2 = 0$
$
\Rightarrow 3{\left( 1 \right)^2} + a\left( 1 \right) - 2 = 0 \\
\Rightarrow 3 + a - 2 = 0 \\
\Rightarrow a + 1 = 0 \\
\Rightarrow a = - 1.....................(1) \\
$
Step 2: now, it is given that given quadratic equation $a({x^2} + 6x) - b = 0$has equal roots. So we can apply the formula (A) for the expression $a({x^2} + 6x) - b = 0$
$ \Rightarrow {\left( {6a} \right)^2} = 4\left( a \right)\left( { - b} \right)$
Now, simply the expression just obtained above,
$ \Rightarrow 36{a^2} = - 4ab..................(2)$
Step 3: Now, we substitute the value of expression (1) in the expression (2) that is obtained in the solution step 2.
$
\Rightarrow 36{\left( { - 1} \right)^2} = - 4\left( { - 1} \right)b \\
\Rightarrow 36 = 4b \\
\Rightarrow b = 9 \\
$
Final solution: Hence, the value of $b$ is 9 when 1 is the root of the quadratic equation $3{x^2} + ax - 2 = 0$ and the quadratic equation $a({x^2} + 6x) - b = 0$ has equal roots.
Note:
First of all we have to put $x = 1$ in the expression $3{x^2} + ax - 2 = 0$ to obtain the value of $a$
We have to use the formula ${b^2} = 4ac$ for the equal roots of the given expression $a({x^2} + 6x) - b = 0$
Now, as mentioned in the question that the quadratic expression $a({x^2} + 6x) - b = 0$ has two equal roots so we have to use the formula for the quadratic expression when the roots are equal to each other which is as mentioned below:
$ \Rightarrow {b^2} = 4ac..............(A)$
Hence, on substituting all the values in the formula (A) just above in the expression $a({x^2} + 6x) - b = 0$ we can determine the value of $b$and we have to also substitute the value of a as obtained with the help of the expression$3{x^2} + ax - 2 = 0$ which we have already obtained.
Complete answer:
Step 1: Since, as given in the question that 1 is the root of the quadratic equation $3{x^2} + ax - 2 = 0$so we have to put $x = 1$in the given expression$3{x^2} + ax - 2 = 0$
$
\Rightarrow 3{\left( 1 \right)^2} + a\left( 1 \right) - 2 = 0 \\
\Rightarrow 3 + a - 2 = 0 \\
\Rightarrow a + 1 = 0 \\
\Rightarrow a = - 1.....................(1) \\
$
Step 2: now, it is given that given quadratic equation $a({x^2} + 6x) - b = 0$has equal roots. So we can apply the formula (A) for the expression $a({x^2} + 6x) - b = 0$
$ \Rightarrow {\left( {6a} \right)^2} = 4\left( a \right)\left( { - b} \right)$
Now, simply the expression just obtained above,
$ \Rightarrow 36{a^2} = - 4ab..................(2)$
Step 3: Now, we substitute the value of expression (1) in the expression (2) that is obtained in the solution step 2.
$
\Rightarrow 36{\left( { - 1} \right)^2} = - 4\left( { - 1} \right)b \\
\Rightarrow 36 = 4b \\
\Rightarrow b = 9 \\
$
Final solution: Hence, the value of $b$ is 9 when 1 is the root of the quadratic equation $3{x^2} + ax - 2 = 0$ and the quadratic equation $a({x^2} + 6x) - b = 0$ has equal roots.
Note:
First of all we have to put $x = 1$ in the expression $3{x^2} + ax - 2 = 0$ to obtain the value of $a$
We have to use the formula ${b^2} = 4ac$ for the equal roots of the given expression $a({x^2} + 6x) - b = 0$
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What constitutes the central nervous system How are class 10 biology CBSE

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

Explain the Treaty of Vienna of 1815 class 10 social science CBSE
