Solving Linear–Quadratic Systems - Algebra 2

1. What is the standard form of a horizontal parabola opening left or right?

   $x=ay^2+by+c$. Standard parabola form with horizontal axis of symmetry.

2. What does $b^2-4ac=0$ mean for the number of intersection points?

   There is $1$ real intersection point (tangent). Zero discriminant means one repeated intersection.

3. What does it mean if a line and a parabola intersect at two points?

   The system has $2$ real solutions. Two intersection points means two solution pairs.

4. What method solves a linear–quadratic system by replacing a variable using the linear equation?

   Substitution. Replace one variable using the linear equation.

5. What are the intersection points of $x=1$ and $y=x^2-1$?

   $(1,0)$. Substitute $x = 1$ into $y = x^2 - 1$.

6. What are the intersection points of $y=-2$ and $x^2+y^2=5$?

   $(1,-2)$ and $(-1,-2)$. Set $y = -2$ in $x^2 + y^2 = 5$.

7. What are the intersection points of $y=x-2$ and $y=x^2-4$?

   $(-1,-3)$ and $(2,0)$. Set $x - 2 = x^2 - 4$ and solve.

8. What is the general form of a linear equation in two variables?

   $Ax+By=C$. Standard form for any line equation.

9. What are the intersection points of $y=x$ and $x^2+y^2=8$?

   $(2,2)$ and $(-2,-2)$. Substitute $y = x$ into circle equation.

10. What must you do after finding $x$-values from a substituted quadratic equation?

    Substitute into the line to find corresponding $y$-values. Back-substitute $x$-values to complete solution pairs.

11. What does the discriminant tell you about real intersections after substitution?

    Sign of $b^2-4ac$ gives $0$, $1$, or $2$ real solutions. Discriminant determines intersection count possibilities.

12. What must every solution to a linear–quadratic system be written as?

    An ordered pair $(x,y)$. Solutions are coordinate points $(x,y)$.

13. What are the intersection points of $y=3x$ and $x^2+y^2=10$?

    $(1,3)$ and $(-1,-3)$. Substitute $y = 3x$ into circle equation.

14. What are the intersection points of $y=x$ and $x^2+y^2=2$?

    $(1,1)$ and $(-1,-1)$. Substitute $y = x$ into circle equation.

15. What are the intersection points of $y=4$ and $x^2+y^2=25$?

    $(3,4)$ and $(-3,4)$. Substitute $y = 4$ into $x^2 + y^2 = 25$.

16. What are the intersection points of $y=0$ and $y=x^2-4$?

    $(2,0)$ and $(-2,0)$. Set $0 = x^2 - 4$ giving $x = ±2$.

17. What are the intersection points of $y=-1$ and $x^2+y^2=5$?

    $(2,-1)$ and $(-2,-1)$. Set $y = -1$ in $x^2 + y^2 = 5$.

18. What are the intersection points of $y=-2x$ and $x^2+y^2=5$?

    $(1,-2)$ and $(-1,2)$. Substitute $y = -2x$ into circle equation.

19. What are the intersection points of $x=0$ and $x^2+y^2=16$?

    $(0,4)$ and $(0,-4)$. Set $x = 0$ in circle equation gives $y = ±4$.

20. Identify the intersection count if substitution gives $(x-2)^2=0$.

    $1$ real solution. Perfect square gives one repeated solution.

21. What is the standard form of a circle centered at $(h,k)$ with radius $r$?

    $(x-h)^2+(y-k)^2=r^2$. Circle equation with center and radius parameters.

22. What are the intersection points of $y=2$ and $x^2+y^2=5$?

    $(1,2)$ and $(-1,2)$. Set $y = 2$ in $x^2 + y^2 = 5$.

23. Identify the intersection count if substitution gives $x^2-9=0$.

    $2$ real solutions. Two distinct solutions: $x = ±3$.

24. Identify the intersection count if substitution gives $x^2+1=0$.

    $0$ real solutions. No real $x$-values exist for this equation.

25. What are the intersection points of $y=-x$ and $y=x^2-2$?

    $(-2,2)$ and $(1,-1)$. Set $-x = x^2 - 2$ and solve quadratic.

26. What are the intersection points of $y=-x$ and $y=x^2$?

    $(0,0)$ and $(-1,1)$. Substitute $y = -x$ into $y = x^2$.

27. What are the intersection points of $y=x+2$ and $y=x^2$?

    $(-1,1)$ and $(2,4)$. Set $x + 2 = x^2$ and solve quadratic.

28. What are the intersection points of $y=-x$ and $x^2+y^2=8$?

    $(2,-2)$ and $(-2,2)$. Substitute $y = -x$ into circle equation.

29. What does it mean if a line does not intersect a circle on the coordinate plane?

    The system has $0$ real solutions. No intersection means no real solution pairs.

30. What are the intersection points of $y=1$ and $x=y^2-2$?

    $(-1,1)$. Substitute $y = 1$ into $x = y^2 - 2$.

31. What is the key algebraic step after substituting a linear equation into a quadratic equation?

    Solve the resulting quadratic in one variable. Use quadratic formula or factoring methods.

32. What are the intersection points of $y=-x+2$ and $y=x^2$?

    $(-2,4)$ and $(1,1)$. Set $-x + 2 = x^2$ and solve quadratic.

33. Identify the correct solution set type if a line intersects a parabola at exactly one point.

    One real solution (the line is tangent to the parabola). Discriminant equals zero for tangent intersections.

34. What is the standard form of a vertical parabola opening up or down?

    $y=ax^2+bx+c$. Standard parabola form with vertical axis of symmetry.

35. What does $b^2-4ac<0$ mean for the number of intersection points?

    There are $0$ real intersection points. Negative discriminant means no real intersections.

36. What is the slope-intercept form of a line used for substitution?

    $y=mx+b$. Isolates $y$ for easy substitution into quadratics.

37. What does it mean if a line is tangent to a circle in a linear–quadratic system?

    The system has exactly $1$ real solution. Tangency creates one point of contact.

38. What is the substituted equation after using $y=7-2x$ in $y=x^2$?

    $x^2=7-2x$. Set the two expressions for $y$ equal.

39. What are the intersection points of $y=2$ and $x=y^2-5$?

    $(-1,2)$. Substitute $y = 2$ into $x = y^2 - 5$.

40. What are the intersection points of $y=2x$ and $x^2+y^2=5$?

    $(1,2)$ and $(-1,-2)$. Substitute $y = 2x$ into circle equation.

41. What are the intersection points of $y=x$ and $y=x^2$?

    $(0,0)$ and $(1,1)$. Substitute $y = x$ into $y = x^2$.

42. What method solves a linear–quadratic system by combining equations to eliminate a variable?

    Elimination (after rewriting equations as needed). Combine equations to cancel out one variable.

43. What are the intersection points of $y=1$ and $x^2+y^2=5$?

    $(2,1)$ and $(-2,1)$. Set $y = 1$ in $x^2 + y^2 = 5$.

44. What should you do first to solve $2x+y=7$ with $y=x^2$ by substitution?

    Rewrite the line as $y=7-2x$. Solve for $y$ to enable substitution method.

45. What are the intersection points of $y=x+1$ and $y=x^2+1$?

    $(0,1)$ and $(1,2)$. Set $x + 1 = x^2 + 1$ and solve.

46. What are the intersection points of $y=0$ and $x^2+y^2=9$?

    $(3,0)$ and $(-3,0)$. Set $y = 0$ in circle equation gives $x = ±3$.

47. What does $b^2-4ac>0$ mean for the number of intersection points?

    There are $2$ real intersection points. Positive discriminant means two distinct intersections.

48. What are the intersection points of $y=3$ and $y=x^2+2$?

    $(1,3)$ and $(-1,3)$. Set $3 = x^2 + 2$ and solve for $x$.

49. What are the intersection points of $y=2x$ and $y=x^2$?

    $(0,0)$ and $(2,4)$. Substitute $y = 2x$ into $y = x^2$.

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