for the following equation, state the value of the discriminate and then describe the nature of the solutions, from the following statements. -20x^2-5x+3=0
20X^2 - 5X + 3 = 0.
D = B^2 - 4AC = 25 - 240 = -215.
The discriminant is negative.Therefore we have 2 imaginary solutions,and no
real solutions.
To find the discriminant ($D$) of the given quadratic equation, you need to use the formula:
$D = b^2 - 4ac$
where $a$, $b$, and $c$ are the coefficients of the equation in the form $ax^2 + bx + c = 0.
Let's apply this formula to the equation -20x^2 - 5x + 3 = 0:
$a = -20$, $b = -5$, and $c = 3$.
Substituting these values, we get:
$D = (-5)^2 - 4(-20)(3)$
$D = 25 + 240$
$D = 265$
Hence, the discriminant of the equation -20x^2 - 5x + 3 = 0 is $D = 265$.
Now, let's analyze the nature of the solutions based on the value of the discriminant:
1. If $D > 0$, then the equation has two distinct real solutions.
2. If $D = 0$, then the equation has one real solution (also known as a double root).
3. If $D < 0$, then the equation has no real solutions (only complex solutions).
In this case, since the discriminant $D = 265$, which is greater than 0, the equation -20x^2 - 5x + 3 = 0 has two distinct real solutions.