he diagram below shows a graph of function y=−5x2+10x+1. Use the graph to determine the value(s) of x when −5x2+10x+1=1

1. Break Down the Problem

We need to find the values of $x$ for which the equation $-5x^2 + 10x + 1 = 1$ holds true. We can simplify this by equating it to zero.

2. Relevant Concepts

To solve for $x$, we will rewrite the equation: $-5x^2 + 10x + 1 - 1 = 0$ This simplifies to: $-5x^2 + 10x = 0$

3. Analysis and Detail

We can factor out common terms from the equation: $-5x(x - 2) = 0$

Setting each factor to zero gives us the potential solutions:

1. $-5x = 0$
2. $x - 2 = 0$

Solving these equations:

1. From $-5x = 0$: $x = 0$
2. From $x - 2 = 0$: $x = 2$

4. Verify and Summarize

Both solutions $x = 0$ and $x = 2$ are derived from setting the equation to zero correctly. To verify, we can substitute these values back into the original equation:

For $x = 0$: $-5(0)^2 + 10(0) + 1 = 1 \quad \text{(True)}$

For $x = 2$: $-5(2)^2 + 10(2) + 1 = -20 + 20 + 1 = 1 \quad \text{(True)}$

Final Answer

The values of $x$ for which $-5x^2 + 10x + 1 = 1$ are: $x = 0 \quad \text{and} \quad x = 2$