PSAT Math Flashcards: Quadratic Equations

What this deck covers

This deck focuses on Quadratic Equations, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

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 discriminant for $ax^2+bx+c=0$?

Answer: $b^2-4ac$. The expression under the square root in the quadratic formula.

Flashcard 2: What is the vertex of $y=a(x-h)^2+k$?

Answer: $(h,k)$. The vertex is the point where the parabola reaches its extreme value.

Flashcard 3: What is the discriminant of $3x^2-6x+1=0$?

Answer: $24$. $\Delta = (-6)^2 - 4(3)(1) = 36 - 12 = 24$.

Flashcard 4: What is the product of the solutions of $ax^2+bx+c=0$ in terms of $a$ and $c$?

Answer: $r_1r_2=\frac{c}{a}$. Another application of Vieta's formulas.

Flashcard 5: What is the factored form of a quadratic with zeros $r_1$ and $r_2$?

Answer: $y=a(x-r_1)(x-r_2)$. Each factor equals zero when $x$ equals a root.

Flashcard 6: What is the vertex $x$-coordinate of $y=ax^2+bx+c$?

Answer: $-\frac{b}{2a}$. The $x$-value where the parabola reaches its maximum or minimum.

Flashcard 7: What condition on $b^2-4ac$ gives no real solutions?

Answer: $b^2-4ac<0$. Negative discriminant means no real square root exists.

Flashcard 8: What condition on $b^2-4ac$ gives exactly two distinct real solutions?

Answer: $b^2-4ac>0$. Positive discriminant means the square root yields two different values.

Flashcard 9: State the quadratic formula for solutions of $ax^2+bx+c=0$.

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general quadratic equation.

Flashcard 10: What is the $y$-intercept of $y=ax^2+bx+c$?

Answer: $(0,c)$. Found by substituting $x=0$ into the equation.

Flashcard 11: Find the axis of symmetry of $y=2x^2-8x+1$.

Answer: $x=2$. Use $x=-\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.

Flashcard 12: What is the vertex form of a quadratic function?

Answer: $y=a(x-h)^2+k$. Shows the vertex at $(h,k)$ and vertical stretch/compression by $a$.

Flashcard 13: What is the direction of opening for $y=ax^2+bx+c$ when $a<0$?

Answer: Opens downward. Negative leading coefficient creates an inverted U-shape.

Flashcard 14: Identify the solutions of $x^2-5x+6=0$.

Answer: $x=2$ and $x=3$. Factor as $(x-2)(x-3)=0$, so $x=2$ or $x=3$.

Flashcard 15: What is the sum of the solutions of $ax^2+bx+c=0$ in terms of $a$ and $b$?

Answer: $r_1+r_2=-\frac{b}{a}$. From Vieta's formulas for quadratic equations.

Flashcard 16: Identify the $y$-intercept of $y=ax^2+bx+c$.

Answer: $(0,c)$. Found by setting $x=0$ in the equation.

Flashcard 17: What condition on $b^2-4ac$ gives exactly one real solution (a double root)?

Answer: $b^2-4ac=0$. Zero discriminant makes $\pm\sqrt{0} = 0$, giving one repeated solution.

Flashcard 18: What does $\Delta<0$ tell you about the solutions of $ax^2+bx+c=0$?

Answer: No real solutions (two complex solutions). Negative under square root means no real solutions exist.

Flashcard 19: What is the discriminant of $ax^2+bx+c=0$?

Answer: $\Delta=b^2-4ac$. The expression under the square root in the quadratic formula.

Flashcard 20: Identify the direction the parabola opens for $y=ax^2+bx+c$ when $a>0$.

Answer: Opens upward. Positive leading coefficient creates a U-shaped parabola.

Flashcard 21: What is the direction of opening for $y=ax^2+bx+c$ when $a>0$?

Answer: Opens upward. Positive leading coefficient creates a U-shaped parabola.

Flashcard 22: Identify the direction the parabola opens for $y=ax^2+bx+c$ when $a<0$.

Answer: Opens downward. Negative leading coefficient creates an inverted U-shape.

Flashcard 23: Solve for $x$: $2x^2+3x-2=0$.

Answer: $x=\frac{1}{2}$ or $x=-2$. Factor as $(2x-1)(x+2)=0$, giving these solutions.

Flashcard 24: What is the axis of symmetry for $y=2x^2-8x+5$?

Answer: $x=2$. Use $x = -\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.

Flashcard 25: Solve for $x$: $x^2-5x+6=0$.

Answer: $x=2$ or $x=3$. Factor as $(x-2)(x-3)=0$, so $x=2$ or $x=3$.

Flashcard 26: What is the standard form of a quadratic equation in one variable?

Answer: $ax^2+bx+c=0$, where $a\ne 0$. Must have $a \ne 0$ to ensure the equation is quadratic, not linear.

Flashcard 27: What is the factored form of a quadratic with zeros $r_1$ and $r_2$?

Answer: $y=a(x-r_1)(x-r_2)$. Each factor equals zero when $x$ equals a root.

Flashcard 28: What is the $x$-coordinate of the vertex for $y=ax^2+bx+c$?

Answer: $x=-\frac{b}{2a}$. Found by completing the square or using calculus to find the minimum/maximum.

Flashcard 29: What is the vertex form of a quadratic function?

Answer: $y=a(x-h)^2+k$. Shows the vertex at $(h,k)$ and vertical stretch/compression by $a$.

Flashcard 30: Identify the solutions of $2x^2+3x-2=0$.

Answer: $x=\frac{1}{2}$ and $x=-2$. Factor as $(2x-1)(x+2)=0$, so $x=\frac{1}{2}$ or $x=-2$.

Flashcard 31: What does $\Delta=0$ tell you about the solutions of $ax^2+bx+c=0$?

Answer: One real solution (a repeated root). Zero discriminant makes $\pm\sqrt{0} = 0$, giving one solution.

Flashcard 32: What is the quadratic formula for solutions to $ax^2+bx+c=0$?

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general quadratic equation.

Flashcard 33: Find the discriminant of $3x^2-4x+1=0$.

Answer: $4$. Calculate $b^2-4ac = (-4)^2-4(3)(1) = 16-12 = 4$.

Flashcard 34: What does $\Delta>0$ tell you about the solutions of $ax^2+bx+c=0$?

Answer: Two distinct real solutions. Positive discriminant means the square root yields two different values.

Flashcard 35: Find the vertex of $y=(x-3)^2-5$.

Answer: $(3,-5)$. In vertex form $y=a(x-h)^2+k$, the vertex is $(h,k)$.

Flashcard 36: What is the standard form of a quadratic equation in one variable?

Answer: $ax^2+bx+c=0$ with $a\ne 0$. Must have $a \ne 0$ to ensure the equation is quadratic, not linear.