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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.

Last updated date: 14th Jun 2024
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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.

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 $a({x^2} + 6x) - b = 0$