Use the discriminant to determine how many real-number solutions the equation has.
v^2 - 7v + 5 = 0
To determine how many real-number solutions the equation has, we can use the discriminant. The discriminant of a quadratic equation in the form ax^2 + bx + c = 0 is given as b^2 - 4ac.
In your equation, v^2 - 7v + 5 = 0, the coefficients are a = 1, b = -7, and c = 5.
So, we can calculate the discriminant as follows:
Discriminant = (-7)^2 - 4(1)(5)
= 49 - 20
= 29
Now, the value of the discriminant is 29.
Here's what it means:
1. If the discriminant is positive, i.e., greater than 0 (in this case, 29 > 0), the equation has two distinct real solutions.
2. If the discriminant is 0 (in this case, not the case), the equation has one repeated real solution (the quadratic touches the x-axis at a single point).
3. If the discriminant is negative (not the case here), the equation has no real solutions, only complex solutions (the quadratic does not intersect the x-axis).
Since the discriminant is positive (greater than 0), the equation v^2 - 7v + 5 = 0 has two distinct real solutions.