Question

Sejam a e b numeros reais nao nulos tais que a equaçao x elevado a 2 + ax + b = 0 possui soluçoes a e b. Determine a-b

Answer 1

$x^2+ax+b=0~~~\Rightarrow~~\left\{ \!\begin{array}{l} A=1\\B=a\\C=b \end{array} \right.$


$\bullet\;\;$ A soma das raízes é $-\,\frac{B}{A}:$

$a+b=-\,\dfrac{a}{1}\\\\\\ a+b=-a\\\\ a+b+a=0\\\\ 2a+b=0~~~~~~\mathbf{(i)}$


$\bullet\;\;$ O produto das raízes é $\frac{C}{A}:$

$a\cdot b=\dfrac{b}{1}\\\\\\ a\cdot b=b~~~~~(\text{mas }b\ne 0)\\\\ a=\dfrac{b}{b}\\\\\\ \boxed{\begin{array}{c}a=1 \end{array}}$


Substituindo em $\mathbf{(i)}\,,$ obtemos

$2\cdot 1+b=0\\\\ 2+b=0\\\\ \boxed{\begin{array}{c} b=-2 \end{array}}$

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Portanto,

$a-b\\\\ =1-(-2)\\\\ =1+2\\\\ =3\\\\\\ \therefore~~\boxed{\begin{array}{c}a-b=3 \end{array}}$