Quadratic EquationHard

Question

Let $a,b$, are real numbers such that $a + b = 5$. Then the equation $x^{2} - ax - b = 0$ must have for all real values of a

Options

A.equal real roots
B.imaginary roots
C.distinct real roots
D.nothing can be said

Solution

$f(x) = x^{2} - ax - b$

and

$$\begin{matrix} f(1) & \ = 1 - a - b = - 4 \\ x & \ \rightarrow \pm \infty,f(x) \rightarrow \infty \end{matrix}$$

$\Rightarrow f(x) = 0$ has two distinct real roots.

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