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Solve the system of equations.\newliney=x2−2x−41y = x^2 - 2x - 41\newliney=−2x−16y = -2x - 16\newline\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)

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Solve a system of linear and quadratic equations: parabolasright chevron icon

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Q. Solve the system of equations.\newliney=x2−2x−41y = x^2 - 2x - 41\newliney=−2x−16y = -2x - 16\newline\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the system of equations:\newliney=x2−2x−41y = x^2 - 2x - 41\newliney=−2x−16y = -2x - 16\newlineTo find the intersection points, we set the two equations equal to each other.\newlinex2−2x−41=−2x−16x^2 - 2x - 41 = -2x - 16
  2. Simplify and Rearrange: Simplify the equation by adding 2x2x and 1616 to both sides to get the equation in standard quadratic form.\newlinex2−2x−41+2x+16=−2x−16+2x+16x^2 - 2x - 41 + 2x + 16 = -2x - 16 + 2x + 16\newlinex2−25=0x^2 - 25 = 0
  3. Solve Quadratic Equation: Solve the quadratic equation for xx.x2−25=0x^2 - 25 = 0This is a difference of squares, which can be factored as:(x−5)(x+5)=0(x - 5)(x + 5) = 0
  4. Factor and Solve: Set each factor equal to zero and solve for xx.(x−5)=0 or (x+5)=0(x - 5) = 0 \text{ or } (x + 5) = 0x=5 or x=−5x = 5 \text{ or } x = -5
  5. Substitute for y: Find the corresponding yy values for each xx by substituting xx back into one of the original equations. We'll use y=−2x−16y = -2x - 16. For x=5x = 5: y=−2(5)−16y = -2(5) - 16 y=−10−16y = -10 - 16 y=−26y = -26
  6. Find yy for x=−5x=-5: Find the yy value for x=−5x = -5:
    y=−2(−5)−16y = -2(-5) - 16
    y=10−16y = 10 - 16
    y=−6y = -6
  7. Write Coordinates: Write the coordinates in exact form.\newlineFirst Coordinate: (5,−26)(5, -26)\newlineSecond Coordinate: (−5,−6)(-5, -6)

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