Question

Find the equation whose roots are $2x_1+3$ and $2x_2+3,$ if $x_1$ and $x_2$ are the roots of $x^2+6x+7=0$.

• A. ${(2x+3)}^2+6(2x+3)+7=0$
• B. ${\left(\frac{x-2}{3}\right)}^2+6\left(\frac{x-2}{3}\right)+7=0$
• C. ${\left(\frac{x-3}{2}\right)}^2+6\left(\frac{x-3}{2}\right)+7=0$
• D. $(2(x-3){)}^2+6[2(x-3)]+7=0$

Answer 1

The correct option is C

${\left(\frac{x-3}{2}\right)}^2+6\left(\frac{x-3}{2}\right)+7=0$

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

$y=2x+3$

$2x+3=y$

$x=\frac{y-3}{2}$

Replace $x$ by $\frac{y-3}{2}$,

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