Question

Find the equation whose roots are \( 2x _1 + 3\) and \(2x _2 + 3,\) if \( x_1\) and \(x _2\) are the roots of \(x^{2} + 6x + 7 = 0\).

Answer 1

We want to find the equation whose roots are $y = 2x + 3$, if $x$ is a root of \(x^{2} + 6x + 7 = 0 \) .

$y = 2x + 3$

$2x + 3 = y$

\( x = \dfrac{y - 3}{2}\)

Replace $x$ by \(\dfrac{y - 3}{2}\),

\(\left ( \dfrac{y-3}{2} \right )^2+6\left ( \dfrac{y-3 }{2} \right )+7=0\\
~~~~~~~~~~~~~~~~~~~~or\\
\left ( \dfrac{x-3}{2} \right )^2+6\left ( \dfrac{x-3}{2} \right )+7=0\)