Question

$x^{2}-34x + c = 0$
in the given equation, $c$ is a constant. the equation has no real solutions if $c>n$. what is the least possible value of $n$?

Explanation:

Step1: Recall discriminant rule for no real roots

For quadratic $ax^2+bx+c=0$, no real solutions when $b^2-4ac<0$.

Step2: Identify coefficients

Here $a=1$, $b=-34$, $c$ is the constant.

Step3: Substitute into discriminant inequality

$$(-34)^2 - 4(1)c < 0$$

Step4: Simplify the inequality

$$1156 - 4c < 0$$
$$-4c < -1156$$
$$c > \frac{1156}{4}$$
$$c > 289$$

Answer:

289