Solve the system of equations. y = x^2 - 14x - 23 y = -14x + 41 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Set Equations Equal: We have the system of equations: y = x^2 - 14x - 23 y = -14x + 41 Set the two equations equal to each other to find the x-values where they intersect. x^2 - 14x - 23 = -14x + 41Simplify and Rearrange: Simplify the equation by adding 14x to both sides and subtracting 41 from both sides to get the quadratic equation in standard form. x^2 - 14x - 23 + 14x - 41 = -14x + 41 + 14x - 41 x^2 - 64 = 0Factor Quadratic Equation: Factor the quadratic equation to find the values of x. x^2 - 64 = (x - 8)(x + 8) Set each factor equal to zero and solve for x. (x - 8) = 0 or (x + 8) = 0 x = 8 or x = -8Solve for x: Find the corresponding y-values for each x-value by substituting back into one of the original equations. We'll use y = -14x + 41. For x = 8: y = -14(8) + 41 y = -112 + 41 y = -71Find y for x=8: Find the corresponding y-value for the second x-value. For x = -8: y = -14(-8) + 41 y = 112 + 41 y = 153Find y for x=-8: Write the coordinates in exact form. The first coordinate is (8, -71). The second coordinate is (-8, 153).

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Solve the system of equations. $\newline$ $y = x^2 - 14x - 23$ $\newline$ $y = -14x + 41$ $\newline$ 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 - 14x - 23

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y = -14x + 41

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = x^2 - 14x - 23$ $\newline$ $y = -14x + 41$ $\newline$ Set the two equations equal to each other to find the $x$ -values where they intersect. $\newline$ $x^2 - 14x - 23 = -14x + 41$
Simplify and Rearrange: Simplify the equation by adding $14x$ to both sides and subtracting $41$ from both sides to get the quadratic equation in standard form. $\newline$ $x^2 - 14x - 23 + 14x - 41 = -14x + 41 + 14x - 41$ $\newline$ $x^2 - 64 = 0$
Factor Quadratic Equation: Factor the quadratic equation to find the values of $x$ . $x^2 - 64 = (x - 8)(x + 8)$ Set each factor equal to zero and solve for $x$ . $(x - 8) = 0$ or $(x + 8) = 0$ $x = 8$ or $x = -8$
Solve for x: Find the corresponding $y$ -values for each $x$ -value by substituting back into one of the original equations. We'll use $y = -14x + 41$ . For $x = 8$ : $y = -14(8) + 41$ $y = -112 + 41$ $y = -71$
Find $y$ for $x=8$ : Find the corresponding $y$ -value for the second $x$ -value. $\newline$ For $x = -8$ : $\newline$ $y = -14(-8) + 41$ $\newline$ $y = 112 + 41$ $\newline$ $y = 153$
Find $y$ for $x=-8$ : Write the coordinates in exact form. $\newline$ The first coordinate is $(8, -71)$ . $\newline$ The second coordinate is $(-8, 153)$ .