How many distinct real solutions does the equation 5x^2 + 5 = 10x have?

How many distinct real solutions does the equation 5x^2 + 5 = 10x have?

• A. 0
• B. 1
• C. 2
• D. 3

Answer: (B)

Quadratic equations can have zero, one, or two real solutions, and the number of distinct real solutions depends on whether the roots are real and whether they are distinct. Start by moving all terms to one side: 5x^2 - 10x + 5 = 0. Factor out 5 to get x^2 - 2x + 1 = 0, which is a perfect square: (x - 1)^2 = 0. This gives x = 1 as a root, and since it comes from a squared factor, it’s a repeated root. Therefore, there is exactly one distinct real solution. The discriminant would be Δ = (-10)^2 - 4·5·5 = 0, confirming there is only one real solution.

Quadratic equations can have zero, one, or two real solutions, and the number of distinct real solutions depends on whether the roots are real and whether they are distinct. Start by moving all terms to one side: 5x^2 - 10x + 5 = 0. Factor out 5 to get x^2 - 2x + 1 = 0, which is a perfect square: (x - 1)^2 = 0. This gives x = 1 as a root, and since it comes from a squared factor, it’s a repeated root. Therefore, there is exactly one distinct real solution. The discriminant would be Δ = (-10)^2 - 4·5·5 = 0, confirming there is only one real solution.