If 1 is a Root of 3x² + ax − 2 = 0 and a(x² + 6x) − b = 0 Has Equal Roots, Find b

Question:

If \(1\) is a root of the quadratic equation

\(3x^2+ax-2=0\)

and the quadratic equation

\(a(x^2+6x)-b=0\)

has equal roots, find the value of \(b\).

Solution

Since \(1\) is a root of \(3x^2+ax-2=0\),

\(3(1)^2+a(1)-2=0\)

\(3+a-2=0\)

\(a+1=0\)

\(a=-1\)

Substituting \(a=-1\) in the second equation,

\(-x^2-6x-b=0\)

\(x^2+6x+b=0\)

For equal roots, the discriminant must be zero.

\(D=b^2-4ac=0\)

\(6^2-4(1)(b)=0\)

\(36-4b=0\)

\(b=9\)

Answer

\( \boxed{b=9} \)