In the equation $ax^2+bx+c=0$, coefficients $a$, $b$, and $c$ are positive real numbers. If $a=c$ and both roots of the quadratic equation are real numbers, which of the following relations must hold between $a$ and $c$ ?

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Since the roots are real numbers, the discriminant cannot be negative. Recall that if the discriminant is negative, the quadratic formula will produce imaginary solutions.

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ $$b^2-4ac \geq 0$$

If $a=c$,

$$b^2-4a(a) \geq 0$$ $$b^2 -4a^2 \geq 0$$ $$b^2 \geq 4a^2$$ $$b \geq 2a$$ $$a \leq \frac{b}{2}$$