Solve the system of equations. y = x^2 - 3x + 19 y = -15x - 13 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Set Equations Equal: We have the system of equations: y = x^2 - 3x + 19 y = -15x - 13 Set the two equations equal to each other to find the x-coordinates of the intersection points. x^2 - 3x + 19 = -15x - 13Rearrange and Form Quadratic Equation: Rearrange the equation to bring all terms to one side and set it equal to zero to form a standard quadratic equation. x^2 - 3x + 19 + 15x + 13 = 0 x^2 + 12x + 32 = 0Factor Quadratic Equation: Factor the quadratic equation. In quadratic equation ax^2 + bx + c, the factors are of the form (x + m)(x + n), where b is the sum and c is the product of m and n respectively. x^2 + 12x + 32 = 0 (x + 4)(x + 8) = 0Solve for x: Solve for x by setting each factor equal to zero. (x + 4) = 0 or (x + 8) = 0 x = -4 or x = -8Find Corresponding y-Values: Find the corresponding y-values by substituting x = -4 and x = -8 into either of the original equations. We'll use y = -15x - 13. For x = -4: y = -15(-4) - 13 = 60 - 13 = 47 For x = -8: y = -15(-8) - 13 = 120 - 13 = 107Write Coordinates: Write the coordinates in exact form. First Coordinate: (-4, 47) Second Coordinate: (-8, 107)

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Solve the system of equations. $\newline$ $y = x^2 - 3x + 19$ $\newline$ $y = -15x - 13$ $\newline$ 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 - 3x + 19

\newline

y = -15x - 13

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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: $\newline$ $y = x^2 - 3x + 19$ $\newline$ $y = -15x - 13$ $\newline$ Set the two equations equal to each other to find the $x$ -coordinates of the intersection points. $\newline$ $x^2 - 3x + 19 = -15x - 13$
Rearrange and Form Quadratic Equation: Rearrange the equation to bring all terms to one side and set it equal to zero to form a standard quadratic equation. $\newline$ $x^2 - 3x + 19 + 15x + 13 = 0$ $\newline$ $x^2 + 12x + 32 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ In quadratic equation $ax^2 + bx + c$ , the factors are of the form $(x + m)(x + n)$ , where $b$ is the sum and $c$ is the product of $m$ and $n$ respectively. $\newline$ $x^2 + 12x + 32 = 0$ $\newline$ $(x + 4)(x + 8) = 0$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $(x + 4) = 0$ or $(x + 8) = 0$ $\newline$ $x = -4$ or $x = -8$
Find Corresponding y-Values: Find the corresponding y-values by substituting $x = -4$ and $x = -8$ into either of the original equations. We'll use $y = -15x - 13$ . For $x = -4$ : $y = -15(-4) - 13 = 60 - 13 = 47$ For $x = -8$ : $y = -15(-8) - 13 = 120 - 13 = 107$
Write Coordinates: Write the coordinates in exact form. $\newline$ First Coordinate: $(-4, 47)$ $\newline$ Second Coordinate: $(-8, 107)$