Quadratic EquationHard
Question
The complete set of real values of 'a' for which there are distinct real numbers $x,y$ satisfying the equations $x = a - y^{2}$ and $y = a - x^{2}$ is
Options
A.$\left\lbrack \frac{3}{4},\infty \right)$
B.$\left\lbrack \frac{5}{4},\infty \right)$
C.$\left\lbrack \frac{7}{4},\infty \right)$
D.$(0,\infty)$
Solution
$x = a - y^{2},\ y = a - x^{2}$
$${\Rightarrow \ x - y = x^{2} - y^{2} \Rightarrow \ x + y = 1 }{\therefore\ 1 - x = a - x^{2} \Rightarrow x^{2} - x + 1 - a = 0 }{D \geq 0\ \Rightarrow \ 1 - 4 + 4a \geq 0\ \Rightarrow \ a \geq \frac{3}{4}}$$
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