The discriminant is a part of the quadratic formula that allows us to determine the nature and number of roots a quadratic equation has. Calculating the discriminant helps to identify how many solutions a quadratic equation has and also assists in determining their type - real or complex.
To find the discriminant, the equation first should be rewritten into its standard form, which is ax^2 + bx + c = 0. Therefore, the equation 4r^2 = 8r - 1 may be rewritten as 4r^2 - 8r + 1 = 0.
In this case, the coefficient a is 4, b is -8 and c is 1.
The formula for the discriminant is D = b^2 - 4ac.
Substituting the relevant values from the equation, we get:
D = (-8)^2 - 4*4*1 = 64 - 16 = 48
Therefore, the discriminant for the given equation is 48.
The discriminant can give us important clues about the roots of the quadratic equation:
- If D > 0, there are two distinct real solutions.
- If D = 0, there's one real root.
- If D < 0, the solutions are complex/imaginary.
Since our calculated discriminant is 48, which is greater than zero, there will be two distinct real solutions for the quadratic equation 4r^2 - 8r + 1 = 0.