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Solve the system of equations. $\newline$ $y = 2x^2 + 39x + 43$ $\newline$ $y = 23x + 11$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,)

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Solve a system of linear and quadratic equations

Full solution

Q. Solve the system of equations.

\newline

y = 2x^2 + 39x + 43

\newline

y = 23x + 11

\newline

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

\newline

(______,______)

Set Equations Equal: Set the two equations equal to each other since they both equal $y$ . $2x^2 + 39x + 43 = 23x + 11$
Subtract and Simplify: Subtract $23x + 11$ from both sides to set the equation to zero. $\newline$ $2x^2 + 39x + 43 - 23x - 11 = 0$ $\newline$ $2x^2 + 16x + 32 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ $2x^2 + 16x + 32 = 0$ $\newline$ $2(x^2 + 8x + 16) = 0$ $\newline$ $2(x + 4)^2 = 0$
Solve for x: Solve for x by setting the factor equal to zero. $\newline$ $(x + 4)^2 = 0$ $\newline$ $x + 4 = 0$ $\newline$ $x = -4$
Find y Value: Substitute $x$ back into one of the original equations to find $y$ . $\newline$ Using $y = 23x + 11$ : $\newline$ $y = 23(-4) + 11$ $\newline$ $y = -92 + 11$ $\newline$ $y = -81$

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