Questions: y=-x^2+2x+5 y=x-1

Transcript text: $\begin{array}{c}y=-x^{2}+2 x+5 \\ y=x-1\end{array}$

Solution

Solution Steps

Step 1: Set Up the Equations

We have two equations: \[ y = -x^2 + 2x + 5 \] \[ y = x - 1 \]

Step 2: Find Points of Intersection

To find the points of intersection, we set the two equations equal to each other: \[ -x^2 + 2x + 5 = x - 1 \] Rearranging gives us: \[ -x^2 + 2x - x + 5 + 1 = 0 \implies -x^2 + x + 6 = 0 \] Multiplying through by -1: \[ x^2 - x - 6 = 0 \]

Step 3: Solve for $ x $

Factoring the quadratic equation: \[ (x - 3)(x + 2) = 0 \] Thus, the solutions for $ x $ are: \[ x = 3 \quad \text{and} \quad x = -2 \]

Step 4: Find Corresponding $ y $ Values

Substituting $ x = 3 $ into $ y = x - 1 $: \[ y = 3 - 1 = 2 \] Substituting $ x = -2 $ into $ y = x - 1 $: \[ y = -2 - 1 = -3 \]

Final Answer

The points of intersection are: \[ \boxed{(-2, -3) \text{ and } (3, 2)} \]

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