Question

3 find the discriminant of each quadratic equation then state the number and type of solutions. \\(-8k^2 + 10k + 3 = 0\\) the discriminant is \\(\boxed{}\\), and there is/are \\(\boxed{}\\) \\(\boxed{}\\) solution(s).

Explanation:

Step1: Identify quadratic coefficients

For $-8k^2 + 10k + 3 = 0$, we have $a=-8$, $b=10$, $c=3$.

Step2: Apply discriminant formula

The discriminant of $ax^2+bx+c=0$ is $\Delta = b^2 - 4ac$.
$\Delta = 10^2 - 4(-8)(3)$

Step3: Calculate the discriminant

$\Delta = 100 + 96 = 196$

Step4: Analyze discriminant

Since $\Delta>0$ and is a perfect square ($196=14^2$), there are 2 distinct real rational solutions.

Answer:

The discriminant is $196$, and there are 2 distinct rational real solution(s).