For which values of A and B will the roots of the equation x^2 + Ax + B = 0 be A and B?
The question is asking to find the value of A and B that will yield 0 = 0.
Is the formula Ax + By + C = 0 used in any way here? Can someone get me started?
For which values of A and B will the roots of the equation x^2 + Ax + B = 0 be A and B?
The question is asking to find the value of A and B that will yield 0 = 0.
Is the formula Ax + By + C = 0 used in any way here? Can someone get me started?
by the quadratic formula the roots of $a x^2 + b x + c$ are
$r_{1,2} = \dfrac{-b \pm \sqrt{b^2 - 4 a c}}{2a}$
$r_1 + r_2 = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2a}+ \dfrac{-b -\sqrt{b^2 - 4 a c}}{2a} = \dfrac {-2b}{2a} = -\dfrac b a$
$r_1 \times r_2 = \dfrac{b^2 -(\sqrt{b^2 - 4ac})^2}{4a^2} = \dfrac{b^2 - (b^2-4ac)}{4a^2} = \dfrac{4 a c}{4 a^2} = \dfrac {c}{a}$
with
$a=1,~b=A,~c=B$ this translates to
$r_1+r_2 = -A$
$r_1 \times r_2 = B$
No, it's not. "0= 0" is true for any values of A and B.
If both A and B satisfy $\displaystyle x^2+ Ax+ B= 0$ then $\displaystyle A^2+ A(A)+ B= 2A^2+ B= 0$, so that $\displaystyle B= -A^2$, and $\displaystyle B^2+ AB+ B= 0$. Since $\displaystyle B= -A^2$, $\displaystyle (-A^2)^2+ A(-A^2)+ (-A^2)= A^4- A^3- A^2= A^2(A^2- A- 1)= 0$.Is the formula Ax + By + C = 0 used in any way here? Can someone get me started?