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 \(a=c\),

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