Question

question 16 (4 points)
determine the points of intersection of the line \\(y = -4x + 7\\) and the curve \\(y = 3x^2 + 5x - 23\\).

Explanation:

Equate the two functions

To find the intersection points, set the linear equation equal to the quadratic equation.

$$ 3x^2 + 5x - 23 = -4x + 7 $$

Rearrange into standard form

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK0
$$

Simplify and solve for x

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK1
$$

Calculate corresponding y-values

Substitute the \(x\)-values back into the linear equation \(y = -4x + 7\) to find the \(y\)-coordinates.

For \(x = -5\):

$$ y = -4(-5) + 7 = 20 + 7 = 27 $$

For \(x = 2\):

$$ y = -4(2) + 7 = -8 + 7 = -1 $$

State the intersection points

The points of intersection are \((-5, 27)\) and \((2, -1)\).

Answer:

The points of intersection are \((-5, 27)\) and \((2, -1)\).