Solve the system of equations. y = x^2 - 2x - 41 y = -2x - 16 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Set Equations Equal: We have the system of equations: y = x^2 - 2x - 41 y = -2x - 16 To find the intersection points, we set the two equations equal to each other. x^2 - 2x - 41 = -2x - 16Simplify and Rearrange: Simplify the equation by adding 2x and 16 to both sides to get the equation in standard quadratic form. x^2 - 2x - 41 + 2x + 16 = -2x - 16 + 2x + 16 x^2 - 25 = 0Solve Quadratic Equation: Solve the quadratic equation for x. x^2 - 25 = 0 This is a difference of squares, which can be factored as: (x - 5)(x + 5) = 0Factor and Solve: Set each factor equal to zero and solve for x. (x - 5) = 0 or (x + 5) = 0 x = 5 or x = -5Substitute for y: Find the corresponding y values for each x by substituting x back into one of the original equations. We'll use y = -2x - 16. For x = 5: y = -2(5) - 16 y = -10 - 16 y = -26Find y for x=-5: Find the y value for x = -5: y = -2(-5) - 16 y = 10 - 16 y = -6Write Coordinates: Write the coordinates in exact form. First Coordinate: (5, -26) Second Coordinate: (-5, -6)

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Solve the system of equations. $\newline$ $y = x^2 - 2x - 41$ $\newline$ $y = -2x - 16$ $\newline$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Algebra 2

Solve a system of linear and quadratic equations: parabolas

Full solution

Q. Solve the system of equations.

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y = x^2 - 2x - 41

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y = -2x - 16

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Write the coordinates in exact form. Simplify all fractions and radicals.

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(______,______)

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(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = x^2 - 2x - 41$ $\newline$ $y = -2x - 16$ $\newline$ To find the intersection points, we set the two equations equal to each other. $\newline$ $x^2 - 2x - 41 = -2x - 16$
Simplify and Rearrange: Simplify the equation by adding $2x$ and $16$ to both sides to get the equation in standard quadratic form. $\newline$ $x^2 - 2x - 41 + 2x + 16 = -2x - 16 + 2x + 16$ $\newline$ $x^2 - 25 = 0$
Solve Quadratic Equation: Solve the quadratic equation for $x$ . $x^2 - 25 = 0$ This is a difference of squares, which can be factored as: $(x - 5)(x + 5) = 0$
Factor and Solve: Set each factor equal to zero and solve for $x$ . $(x - 5) = 0 \text{ or } (x + 5) = 0$ $x = 5 \text{ or } x = -5$
Substitute for y: Find the corresponding $y$ values for each $x$ by substituting $x$ back into one of the original equations. We'll use $y = -2x - 16$ . For $x = 5$ : $y = -2(5) - 16$ $y = -10 - 16$ $y = -26$
Find $y$ for $x=-5$ : Find the $y$ value for $x = -5$ :
$y = -2(-5) - 16$
$y = 10 - 16$
$y = -6$
Write Coordinates: Write the coordinates in exact form. $\newline$ First Coordinate: $(5, -26)$ $\newline$ Second Coordinate: $(-5, -6)$