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 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 2: 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 3: 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 4: 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 5: What are the intersection points of $x=1$ and $y=x^2-1$?

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

Flashcard 6: 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 7: 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 8: What is the general form of a linear equation in two variables?

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

Flashcard 9: 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 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 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 12: 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 13: What are the intersection points of $y=3x$ and $x^2+y^2=10$?

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

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

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

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

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

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

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

Flashcard 17: 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 18: What are the intersection points of $y=-2x$ and $x^2+y^2=5$?

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

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

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

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

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

Flashcard 21: 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 22: 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 23: Identify the intersection count if substitution gives $x^2-9=0$.

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

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

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

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

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

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

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

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

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

Flashcard 28: 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 29: 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 30: What are the intersection points of $y=1$ and $x=y^2-2$?

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

Flashcard 31: 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 32: What are the intersection points of $y=-x+2$ and $y=x^2$?

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

Flashcard 33: 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 34: 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 35: What does $b^2-4ac<0$ mean for the number of intersection points?

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

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

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

Flashcard 37: 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 38: 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 39: What are the intersection points of $y=2$ and $x=y^2-5$?

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

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

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

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

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

Flashcard 42: 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 43: 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 44: 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 45: What are the intersection points of $y=x+1$ and $y=x^2+1$?

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

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

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

Flashcard 47: 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 48: What are the intersection points of $y=3$ and $y=x^2+2$?

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

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

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