Final answer:
The equation 25x² − 20x + 4 = 0 has one real, repeated solution, which is x = 0.4, determined by using the discriminant method which results in a value of 0.
Step-by-step explanation:
To identify the number and type of solutions for the quadratic equation 25x² − 20x + 4 = 0, we can use the discriminant method. The discriminant is given by the formula b² - 4ac, where the coefficients a, b, and c correspond to the terms in the standard form of a quadratic equation ax² + bx + c = 0. In this equation, a is 25, b is -20, and c is 4.
Now, let's calculate the discriminant:
(-20)² - 4(25)(4) = 400 - 400 = 0
A discriminant of 0 indicates that the quadratic equation has one unique solution, and the graph of the equation will touch the x-axis at one point. The equation is a perfect square, which can be factored as (5x - 2)². Therefore, the solution for the equation is the value of x for which 5x - 2 = 0, which gives us x = 0.4.
The number of solutions in this case is one, and it is a real and repeated solution since both solutions are the same.