Solve the system of equations. y = -27x + 24 y = 12x^2 - 27x - 24 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Set Equations Equal: Set the two equations equal to each other to find the x-values where they intersect. y = -27x + 24 y = 12x^2 - 27x - 24 -27x + 24 = 12x^2 - 27x - 24Simplify Equation: Simplify the equation by moving all terms to one side to form a standard quadratic equation. -27x + 24 - (-27x + 24) = 12x^2 - 27x - 24 - (-27x + 24) 0 = 12x^2 - 48Divide and Simplify: Divide the equation by 12 to simplify it further. 0 = 12x^2 - 48 0 = x^2 - 4Factor Quadratic Equation: Factor the quadratic equation. x^2 - 4 = (x - 2)(x + 2)Solve for x: Solve for x by setting each factor equal to zero. (x - 2) = 0 or (x + 2) = 0 x = 2 or x = -2Substitute and Find y: Substitute the x-values back into one of the original equations to find the corresponding y-values. For x = 2: y = -27(2) + 24 y = -54 + 24 y = -30 For x = -2: y = -27(-2) + 24 y = 54 + 24 y = 78Write Coordinates: Write the coordinates in exact form. First Coordinate: (2, -30) Second Coordinate: (-2, 78)

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Solve the system of equations. $\newline$ $y = -27x + 24$ $\newline$ $y = 12x^2 - 27x - 24$ $\newline$ 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

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Q. Solve the system of equations.

\newline

y = -27x + 24

\newline

y = 12x^2 - 27x - 24

\newline

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: Set the two equations equal to each other to find the $x$ -values where they intersect. $y = -27x + 24$ $y = 12x^2 - 27x - 24$ $-27x + 24 = 12x^2 - 27x - 24$
Simplify Equation: Simplify the equation by moving all terms to one side to form a standard quadratic equation. $\newline$ $-27x + 24 - (-27x + 24) = 12x^2 - 27x - 24 - (-27x + 24)$ $\newline$ $0 = 12x^2 - 48$
Divide and Simplify: Divide the equation by $12$ to simplify it further. $\newline$ $0 = 12x^2 - 48$ $\newline$ $0 = x^2 - 4$
Factor Quadratic Equation: Factor the quadratic equation. $x^2 - 4 = (x - 2)(x + 2)$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $(x - 2) = 0$ or $(x + 2) = 0$ $\newline$ $x = 2$ or $x = -2$
Substitute and Find $y$ : Substitute the $x$ -values back into one of the original equations to find the corresponding $y$ -values. $\newline$ For $x = 2$ : $\newline$ $y = -27(2) + 24$ $\newline$ $y = -54 + 24$ $\newline$ $y = -30$ $\newline$ For $x = -2$ : $\newline$ $y = -27(-2) + 24$ $\newline$ $y = 54 + 24$ $\newline$ $x$ $0$
Write Coordinates: Write the coordinates in exact form. $\newline$ First Coordinate: $(2, -30)$ $\newline$ Second Coordinate: $(-2, 78)$