GRE Quantitative Flashcards: Quadratic Polynomial Relationships

What this deck covers

This deck focuses on Quadratic Polynomial Relationships, giving you a quick way to review the definitions, rules, and examples that matter most for GRE Quantitative.

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 vertex of $y=ax^2+bx+c$ written in terms of $a,b,c$?

Answer: $\left(-\frac{b}{2a},\frac{4ac-b^2}{4a}\right)$. The vertex coordinates are found by using the axis of symmetry for $x$ and substituting to find $y$, yielding the given expressions in terms of $a$, $b$, $c$.

Flashcard 2: What is the axis of symmetry of $y=ax^2+bx+c$?

Answer: $x=-\frac{b}{2a}$. The axis of symmetry for a parabola $y = ax^2 + bx + c$ is the vertical line through the vertex at $x = -b/(2a)$.

Flashcard 3: Solve $x^2-5x+6=0$.

Answer: $x=2,3$. Factoring gives $(x-2)(x-3)=0$, so roots are $2$ and $3$.

Flashcard 4: What is the relationship between roots and coefficients for a monic quadratic $x^2+px+q$?

Answer: Roots sum $=-p$, product $=q$. For monic quadratics, Vieta's formulas simplify to sum of roots $=-p$ and product $=q$.

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

Answer: $a(x-r_1)(x-r_2)$. The factored form arises from the roots being the values that make the quadratic zero, scaled by the leading coefficient $a$.

Flashcard 6: What is the quadratic formula for solutions of $ax^2+bx+c=0$?

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. The quadratic formula derives from completing the square to solve $ax^2 + bx + c = 0$ for $x$.

Flashcard 7: What is the discriminant of $ax^2+bx+c$ and what does its sign determine?

Answer: $\Delta=b^2-4ac$; sign gives number of real roots. The discriminant $Delta = b^2 - 4ac$ determines the nature of roots: positive for two real, zero for one real, negative for no real roots.

Flashcard 8: Find the quadratic with roots $2$ and $-3$ and leading coefficient $1$.

Answer: $x^2+x-6$. Using sum $2 + (-3) = -1$ and product $2 imes (-3) = -6$ in monic form.

Flashcard 9: What is the axis of symmetry of $y=5x^2-20x+1$?

Answer: $x=2$. The axis of symmetry is $x = -b/(2a) = 20/(10) = 2$.

Flashcard 10: Factor $x^2-9x+20$ over the integers.

Answer: $(x-5)(x-4)$. Factors are found by pairs that multiply to $20$ and add to $-9$, yielding $-4$ and $-5$.

Flashcard 11: Identify the discriminant of $2x^2+3x-7$.

Answer: $\Delta=65$. The discriminant is calculated as $b^2 - 4ac = 9 - 4(2)(-7) = 9 + 56 = 65$.

Flashcard 12: Find the sum and product of roots of $3x^2-12x+5=0$.

Answer: Sum $=4$, product $=\frac{5}{3}$. Applying Vieta's formulas to $3x^2 - 12x + 5 = 0$ gives sum $=12/3=4$ and product $=5/3$.

Flashcard 13: What are the sum and product of roots of $ax^2+bx+c=0$ in terms of $a,b,c$?

Answer: Sum $=-\frac{b}{a}$, product $=\frac{c}{a}$. Vieta's formulas state that for $ax^2 + bx + c = 0$, the sum of the roots is $-b/a$ and the product is $c/a$.

Flashcard 14: Identify the number of real roots of $ax^2+bx+c=0$ when $b^2-4ac>0$.

Answer: Two distinct real roots. A positive discriminant indicates the quadratic crosses the x-axis at two distinct points.

Flashcard 15: What is the vertex form of a quadratic, and what is the vertex in that form?

Answer: $y=a(x-h)^2+k$; vertex $=(h,k)$. Vertex form is obtained by completing the square, with $(h, k)$ representing the vertex's coordinates.

Flashcard 16: What is the minimum or maximum value of $y=a(x-h)^2+k$ in terms of $a$ and $k$?

Answer: If $a>0$, min $=k$; if $a<0$, max $=k$. In vertex form, $k$ is the extremum value, minimum if $a>0$ and maximum if $a<0$.

Flashcard 17: If roots of $x^2-kx+12=0$ are $3$ and $4$, what is $k$?

Answer: $k=7$. Sum of roots $3+4=7$ equals $k$ for the quadratic $x^2 - kx +12=0$.

Flashcard 18: Identify the number of real roots of $ax^2+bx+c=0$ when $b^2-4ac=0$.

Answer: One real double root. A zero discriminant means the quadratic touches the x-axis at exactly one point, a repeated real root.

Flashcard 19: What is the $y$-intercept of $y=ax^2+bx+c$?

Answer: $c$. The $y$-intercept occurs at $x=0$, so substituting gives $y=c$.

Flashcard 20: What is the quadratic whose roots are $r$ and $s$ and whose leading coefficient is $1$?

Answer: $x^2-(r+s)x+rs$. The monic quadratic is constructed using Vieta's formulas with sum $r+s$ and product $rs$.

Flashcard 21: Identify the number of real roots of $ax^2+bx+c=0$ when $b^2-4ac<0$.

Answer: No real roots (two complex conjugates). A negative discriminant indicates no real solutions, resulting in two complex conjugate roots.

Flashcard 22: What is the condition on $a$ for $y=ax^2+bx+c$ to open upward versus downward?

Answer: Upward if $a>0$; downward if $a<0$. The sign of $a$ determines the parabola's direction: positive $a$ opens upward, negative $a$ opens downward.

Flashcard 23: Find the vertex of $y=x^2-6x+11$.

Answer: $(3,2)$. The vertex is at $x=3$, and $y= (3)^2 -6(3) +11=2$, so $(3,2)$.