Determine where the relation y = -2x² + 16x - 23 intersects the line y = 2x - 3.

Steps to Solve

1. Set the equations equal to each other

To find the intersection points, we need to set the quadratic equation equal to the linear equation: $$ -2x^2 + 16x - 23 = 2x - 3 $$

2. Rearrange the equation

Rearranging the equation will help us bring all terms to one side: $$ -2x^2 + 16x - 23 - 2x + 3 = 0 $$ This simplifies to: $$ -2x^2 + 14x - 20 = 0 $$

3. Factor the quadratic equation

Next, we factor the equation. Dividing through by -2 gives: $$ x^2 - 7x + 10 = 0 $$ Now we can factor this as: $$ (x - 5)(x - 2) = 0 $$

4. Solve for x values

Setting each factor equal to zero gives us: $$ x - 5 = 0 \quad \Rightarrow \quad x = 5 $$ $$ x - 2 = 0 \quad \Rightarrow \quad x = 2 $$

5. Substitute x values back to find y values

Now, we substitute back into the linear equation to find the y-values. Using ( y = 2x - 3 ):

• For ( x = 5 ): $$ y = 2(5) - 3 = 10 - 3 = 7 $$
• For ( x = 2 ): $$ y = 2(2) - 3 = 4 - 3 = 1 $$

6. Identify the intersection points

The intersection points are ( (5, 7) ) and ( (2, 1) ).