Algebra 2 Flashcards: Solving Linear Quadratic Systems

What this deck covers

This deck focuses on Solving Linear Quadratic Systems, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What are the intersection points of $y=2x$ and $y=x^2$?

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

Flashcard 2: What method solves a linear–quadratic system by combining equations to eliminate a variable?

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

Flashcard 3: What is the standard form of a horizontal parabola opening left or right?

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

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

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

Flashcard 5: What is the standard form of a vertical parabola opening up or down?

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

Flashcard 6: What does $b^2-4ac=0$ mean for the number of intersection points?

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

Flashcard 7: What does $b^2-4ac>0$ mean for the number of intersection points?

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

Flashcard 8: What does the discriminant tell you about real intersections after substitution?

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

Flashcard 9: What must every solution to a linear–quadratic system be written as?

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

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

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

Flashcard 11: What is the key algebraic step after substituting a linear equation into a quadratic equation?

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

Flashcard 12: What is the standard form of a circle centered at $(h,k)$ with radius $r$?

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

Flashcard 13: What is the standard form of a horizontal parabola opening left or right?

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

Flashcard 14: What is the substituted equation after using $y=7-2x$ in $y=x^2$?

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

Flashcard 15: What should you do first to solve $2x+y=7$ with $y=x^2$ by substitution?

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

Flashcard 16: What method solves a linear–quadratic system by replacing a variable using the linear equation?

Answer: Substitution. Replace one variable using the linear equation.

Flashcard 17: What method solves a linear–quadratic system by combining equations to eliminate a variable?

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

Flashcard 18: What does it mean if a line and a parabola intersect at two points?

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

Flashcard 19: What does it mean if a line is tangent to a circle in a linear–quadratic system?

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

Flashcard 20: What does it mean if a line does not intersect a circle on the coordinate plane?

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

Flashcard 21: What is the general form of a linear equation in two variables?

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

Flashcard 22: What is the slope-intercept form of a line used for substitution?

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

Flashcard 23: Identify the correct solution set type if a line intersects a parabola at exactly one point.

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

Flashcard 24: What are the intersection points of $y=-2$ and $x^2+y^2=5$?

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

Flashcard 25: What are the intersection points of $y=2$ and $x^2+y^2=5$?

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

Flashcard 26: What are the intersection points of $y=-1$ and $x^2+y^2=5$?

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

Flashcard 27: What are the intersection points of $y=1$ and $x^2+y^2=5$?

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

Flashcard 28: What is the standard form of a vertical parabola opening up or down?

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

Flashcard 29: What are the intersection points of $y=-x$ and $x^2+y^2=8$?

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

Flashcard 30: What are the intersection points of $y=x$ and $x^2+y^2=8$?

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