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Solve the system of equations. $\newline$ $y = 14x - 26$ $\newline$ $y = 2x^2 + 14x - 34$ $\newline$ 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 = 14x - 26

\newline

y = 2x^2 + 14x - 34

\newline

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = 14x - 26$ $\newline$ $y = 2x^2 + 14x - 34$ $\newline$ To find the intersection points, we set the two equations equal to each other. $\newline$ $14x - 26 = 2x^2 + 14x - 34$
Simplify Equation: Simplify the equation by subtracting $14x$ from both sides and adding $26$ to both sides. $\newline$ $0 = 2x^2 + 14x - 34 - 14x + 26$ $\newline$ $0 = 2x^2 - 8$
Divide and Simplify: Divide the entire equation by $2$ to simplify it further. $\newline$ $0 = x^2 - 4$
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 $x = -2$
Find y-values: Find the corresponding y-values for each $x$ -value by substituting back into one of the original equations. We'll use $y = 14x - 26$ . For $x = 2$ : $y = 14(2) - 26$ $y = 28 - 26$ $y = 2$
Coordinate Calculation: For $x = -2$ : $\newline$ $y = 14(-2) - 26$ $\newline$ $y = -28 - 26$ $\newline$ $y = -54$
Coordinate Calculation: For $x = -2$ : $y = 14(-2) - 26$ $y = -28 - 26$ $y = -54$ Write the coordinates in exact form. $\text{First Coordinate: } (2, 2)$ $\text{Second Coordinate: } (-2, -54)$

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