If $A$ and $B$ are the root of the equation $3x^2-4x-9=0$, what is the equation whose roots are $(A+3)/(A-3)$ and $(B+3)/(B-3)$
How do I form this equation?
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0the latter one, sorry – 2012-07-11
4 Answers
Recall that if $\alpha, \beta$ are roots of $ax^2 + bx + c$, then we have $\alpha + \beta = -\dfrac{b}a$ and $\alpha \beta = \dfrac{c}a$.
Let $\alpha = \dfrac{A+3}{A-3}$ and $\beta = \dfrac{B+3}{B-3}$.
Then we get that $\alpha + \beta = \dfrac{A+3}{A-3} + \dfrac{B+3}{B-3} = \dfrac{AB + 3B - 3A - 9 + AB + 3A - 3B - 9}{AB - 3(A+B) + 9} = 2\dfrac{AB - 9}{AB - 3(A+B) + 9}$
Similarly, $\alpha \beta = \dfrac{AB + 3(A+B) + 9}{AB - 3(A+B) + 9}$ But since $A$ and $B$ are roots of $3x^2 -4x - 9$, we have that $A+B = \dfrac43$ and $AB = -3$.
Can you finish it off from here?
Move the mouse over the gray area for the complete solution.
$\alpha + \beta = 2 \left(\dfrac{AB - 9}{AB - 3(A+B) + 9} \right) = 2 \times \dfrac{-3-9}{-3 - 3(4/3) + 9} = 2 \times \dfrac{-12}{2} = -12$ $\alpha \beta = \dfrac{-3 + 3 (4/3) + 9}{-3 - 3 (4/3) + 9} = \dfrac{10}2 = 5$ Hence, the desired quadratic polynomial is $y^2 + 12y + 5.$
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0Yes thanks for helping – 2012-07-11
Remember that the quadratic polynomial with roots $r$ and $s$ is $(x-r)(x-s) = x^2-(r+s)x + rs$.
You know that $(x-A)(x-B) = x^2 - \frac{4}{3}x - 3$, so $AB =-3$ and $A+B = \frac{4}{3}$. From these, you should be able to find a quadratic with the roots you want by expressing the fum and the product of $\frac{A+3}{A-3} \qquad\text{and}\qquad \frac{B+3}{B-3} $ in terms of $A+B$ and $AB$.
More generally, suppose $p(x)$ is a polynomial with roots $r_j$. Note that $y = \dfrac{ax+b}{cx+d}$ (where $ad - bc \ne 0$) when $x = \dfrac{dy - b}{-cy + a}$. So $\dfrac{a r_j+b}{c r_j+d}$ (assuming the denominators are nonzero) are roots of $p\left( \dfrac{dy-b}{-cy+a} \right)$. In this case with $p(x) = 3 x^2 - 4 x - 9$ and $a=1$, $b=3$, $c=1$, $d=-3$, $p\left( \dfrac{-3 y - 3}{-y + 1}\right) = 3{\frac { \left( -3\,y-3 \right) ^{2}}{ \left( -y+1 \right) ^{2}}}-4 \,{\frac {-3\,y-3}{-y+1}}-9 = 6\,{\frac {{y}^{2}+12\,y+5}{ \left( y-1 \right) ^{2}}}$ and the roots of this are the roots of $y^2+12y+5$. This method works with polynomials of arbitrary degree, not just quadratics.
A+B = 4/3
AB = -3
(A+3)/(A-3) + (B+3)/(B-3) = 2(AB - 9)/[AB - 3(A+B) + 9]
= 2(-3 - 9)/[-3 - 3(4/3) + 9] = -12
[(A+3)/(A-3)][(B+3)/(B-3)] = [(A+3)(B+3)]/[(A-3)(B-3)] = [AB + 3(A+B) + 9]/[AB - 3(A+B) + 9]
= [-3 + 3(4/3) + 9]/[-3 -3(4/3) + 9]
= [-3 + 4 + 9]/[-3 - 4 + 9]
= 10/2
= 5
The new equation is framed below.
x^2 - (-12)x + (5) = 0
x^2 + 12x + 5 = 0
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0A second degree equation involving just one variable x with roots a and b is framed as follows. x^2 - (a + b)x + (a)(b) = 0 – 2012-07-11