state the value of the discriminant and the discribe the nature of the solution 20x^2-8x-17=0
The discriminant is
b^2 -4ac = 8^2 - 4*(-17)*20
= 64 + 1360 = 1424
The fact that is is positive and not zero implies the equation has two real solutions. They are both irrational, since 1424 is not a perfect square.
To find the value of the discriminant and determine the nature of the solution, we can use the quadratic formula. The quadratic equation you provided is 20x^2 - 8x - 17 = 0.
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, a = 20, b = -8, and c = -17. Plugging these values into the quadratic formula, we get:
x = (-(-8) ± √((-8)^2 - 4 * 20 * (-17))) / (2 * 20)
Simplifying further, we have:
x = (8 ± √(64 + 1360)) / 40
x = (8 ± √(1424)) / 40
Now, let's evaluate the discriminant, which is the expression under the square root in the quadratic formula. For the given equation, the discriminant is:
D = b^2 - 4ac = (-8)^2 - 4 * 20 * (-17)
D = 64 + 1360
D = 1424
So, the value of the discriminant is 1424.
Now, let's analyze the nature of the solutions based on the discriminant value:
1. If the discriminant (D) is greater than 0 (D > 0), then the quadratic equation has two distinct real solutions.
2. If the discriminant is equal to 0 (D = 0), then the quadratic equation has one real solution (a perfect square).
3. If the discriminant is less than 0 (D < 0), then the quadratic equation has no real solutions. The solutions would be complex conjugates.
In our case, the discriminant is 1424, which is greater than 0. Therefore, the quadratic equation 20x^2 - 8x - 17 = 0 has two distinct real solutions.