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Solve the system of equations. $\newline$ $y = x^2 - 43x - 14$ $\newline$ $y = -45x + 34$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Solve a system of linear and quadratic equations

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Q. Solve the system of equations.

\newline

y = x^2 - 43x - 14

\newline

y = -45x + 34

\newline

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = x^2 - 43x - 14$ $\newline$ $y = -45x + 34$ $\newline$ To find the intersection points, set the two equations equal to each other. $\newline$ $x^2 - 43x - 14 = -45x + 34$
Form Quadratic Equation: Bring all terms to one side to form a quadratic equation. $\newline$ $x^2 - 43x - 14 + 45x - 34 = 0$ $\newline$ $x^2 + 2x - 48 = 0$
Factor Quadratic: Factor the quadratic equation. $\newline$ In quadratic equation $ax^2 + bx + c$ , the factors are of the form $(x + m)(x + n)$ . $\newline$ Where $b$ is the sum and $c$ is the product of $m$ and $n$ respectively. $\newline$ $x^2 + 2x - 48 = 0$ $\newline$ $(x + 8)(x - 6) = 0$
Solve for x: Solve for x. $\newline$ Set each factor equal to zero, and solve for x. $\newline$ $(x + 8) = 0$ or $(x - 6) = 0$ $\newline$ $x = -8$ or $x = 6$
Find y-values: Find the corresponding y-values for each x-value by substituting back into either of the original equations. We'll use $y = -45x + 34$ . $\newline$ For $x = -8$ : $\newline$ $y = -45(-8) + 34$ $\newline$ $y = 360 + 34$ $\newline$ $y = 394$
Write Coordinates: For $x = 6$ : $\newline$ $y = -45(6) + 34$ $\newline$ $y = -270 + 34$ $\newline$ $y = \(-236\)
Write Coordinates: For \(x = 6\):\[y = -45(6) + 34\]\[y = -270 + 34\]\[y = -236\]Write the coordinates in exact form.\[\text{First Coordinate: } (-8, 394)\]\[\text{Second Coordinate: } (6, -236)\]

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