Solve the system of equations. y = x^2 - 6x - 27 y = -6x + 9 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Set Equations Equal: We have the system of equations: y = x^2 - 6x - 27 y = -6x + 9 Set the two equations equal to each other to find the x-values where they intersect. x^2 - 6x - 27 = -6x + 9Simplify and Rearrange: Simplify the equation by adding 6x to both sides and subtracting 9 from both sides to get the quadratic equation in standard form. x^2 - 6x - 27 + 6x - 9 = -6x + 9 + 6x - 9 x^2 - 36 = 0Factor Quadratic Equation: Factor the quadratic equation. x^2 - 36 = (x - 6)(x + 6) Set each factor equal to zero to solve for x. (x - 6) = 0 or (x + 6) = 0 x = 6 or x = -6Find y-values: Find the corresponding y-values for each x-value by substituting x back into one of the original equations. We can use y = -6x + 9. For x = 6: y = -6(6) + 9 y = -36 + 9 y = -27Substitute x-values: Find the corresponding y-value for x = -6: y = -6(-6) + 9 y = 36 + 9 y = 45Write Coordinates: Write the coordinates in exact form. The first coordinate is (6, -27). The second coordinate is (-6, 45).

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Solve the system of equations. $\newline$ $y = x^2 - 6x - 27$ $\newline$ $y = -6x + 9$ $\newline$ 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 = x^2 - 6x - 27

\newline

y = -6x + 9

\newline

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = x^2 - 6x - 27$ $\newline$ $y = -6x + 9$ $\newline$ Set the two equations equal to each other to find the $x$ -values where they intersect. $\newline$ $x^2 - 6x - 27 = -6x + 9$
Simplify and Rearrange: Simplify the equation by adding $6x$ to both sides and subtracting $9$ from both sides to get the quadratic equation in standard form. $\newline$ $x^2 - 6x - 27 + 6x - 9 = -6x + 9 + 6x - 9$ $\newline$ $x^2 - 36 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ $x^2 - 36 = (x - 6)(x + 6)$ $\newline$ Set each factor equal to zero to solve for $x$ . $\newline$ $(x - 6) = 0$ or $(x + 6) = 0$ $\newline$ $x = 6$ or $x = -6$
Find y-values: Find the corresponding y-values for each $x$ -value by substituting $x$ back into one of the original equations. We can use $y = -6x + 9$ . $\newline$ For $x = 6$ : $\newline$ $y = -6(6) + 9$ $\newline$ $y = -36 + 9$ $\newline$ $y = -27$
Substitute x-values: Find the corresponding y-value for $x = -6$ : $\newline$ $y = -6(-6) + 9$ $\newline$ $y = 36 + 9$ $\newline$ $y = 45$
Write Coordinates: Write the coordinates in exact form. $\newline$ The first coordinate is $(6, -27)$ . $\newline$ The second coordinate is $(-6, 45)$ .