SAT 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 SAT 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: Find the vertex of $y = -x^2 + 4x - 3$.

Answer: Vertex: $(2, 1)$. Use $x = -\frac{b}{2a} = 2$, then $y = -(4) + 8 - 3 = 1$.

Flashcard 2: Factor $x^2 - 9$.

Answer: $(x - 3)(x + 3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

Flashcard 3: What is the quadratic formula?

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

Flashcard 4: What is the minimum value of $y = x^2 - 6x + 9$?

Answer: Minimum: $0$. This is $(x-3)^2$, so minimum occurs at vertex $(3,0)$.

Flashcard 5: How do you determine the number of real roots using the discriminant?

Answer: If $b^2 - 4ac > 0$, 2 real roots; $=0$, 1 real root; $<0$, no real roots. The discriminant determines whether roots are real or complex.

Flashcard 6: What is the standard form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. General form where $a \neq 0$ determines parabola shape.

Flashcard 7: What is the minimum value of $y = x^2 - 6x + 9$?

Answer: Minimum: $0$. This is $(x-3)^2$, so minimum occurs at vertex $(3,0)$.

Flashcard 8: State the quadratic formula.

Answer: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Solves any quadratic by substituting coefficients $a$, $b$, and $c$.

Flashcard 9: Write $x^2 + 8x + 16$ as a square.

Answer: $(x + 4)^2$. Perfect square trinomial with $a = x$, $b = 4$.

Flashcard 10: Solve for $x$: $x^2 + 2x - 8 = 0$.

Answer: $x = 2, x = -4$. Factor as $(x-2)(x+4) = 0$ or use the quadratic formula.

Flashcard 11: What is the discriminant of $ax^2 + bx + c = 0$?

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

Flashcard 12: Factor $x^2 - 9$.

Answer: $(x - 3)(x + 3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

Flashcard 13: For $y = x^2 + 4x + 4$, what is the vertex?

Answer: Vertex: $(-2, 0)$. This is $(x+2)^2$, so vertex is at $x = -2$, $y = 0$.

Flashcard 14: What is the product of the roots of $ax^2 + bx + c = 0$?

Answer: $\frac{c}{a}$. By Vieta's formulas relating coefficients to roots.

Flashcard 15: Determine the parabola direction for $y = 5x^2 - 3x + 2$.

Answer: Opens upwards. Positive coefficient of $x^2$ means parabola opens upward.

Flashcard 16: Convert $y = x^2 - 4x + 4$ to vertex form.

Answer: $y = (x-2)^2$. This is $(x-2)^2$ expanded, so vertex form shows vertex $(2,0)$.

Flashcard 17: For $y = x^2 + 4x + 4$, what is the vertex?

Answer: Vertex: $(-2, 0)$. This is $(x+2)^2$, so vertex is at $x = -2$, $y = 0$.

Flashcard 18: How do you determine the number of real roots using the discriminant?

Answer: If $b^2 - 4ac > 0$, 2 real roots; $=0$, 1 real root; $<0$, no real roots. The discriminant determines whether roots are real or complex.

Flashcard 19: Identify the axis of symmetry for $y = ax^2 + bx + c$.

Answer: $x = \frac{-b}{2a}$. Derived from completing the square or using calculus.

Flashcard 20: What is the quadratic formula?

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

Flashcard 21: What type of parabola does $y = -x^2$ represent?

Answer: A downward-opening parabola. Negative coefficient of $x^2$ makes the parabola open downward.

Flashcard 22: Simplify: $(x + 2)^2$.

Answer: $x^2 + 4x + 4$. Apply the perfect square formula $(a+b)^2 = a^2 + 2ab + b^2$.

Flashcard 23: Convert $y = 2(x-3)^2 + 4$ to standard form.

Answer: $y = 2x^2 - 12x + 22$. Expand $(x-3)^2 = x^2 - 6x + 9$, then distribute and combine.

Flashcard 24: Convert $y = x^2 + 6x + 8$ to vertex form.

Answer: $y = (x+3)^2 - 1$. Complete the square: $(x+3)^2 = x^2 + 6x + 9$, so subtract 1.

Flashcard 25: What is the vertex form of a quadratic equation?

Answer: $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical shift directly.

Flashcard 26: Identify the axis of symmetry for $y = ax^2 + bx + c$.

Answer: $x = \frac{-b}{2a}$. Derived from completing the square or using calculus.

Flashcard 27: What is the standard form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. General form where $a \neq 0$ determines parabola shape.

Flashcard 28: What is the standard form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. The general form where $a \neq 0$ defines a parabola.

Flashcard 29: What is the vertex of $y = 2(x-1)^2 + 3$?

Answer: Vertex: $(1, 3)$. In vertex form $y = a(x-h)^2 + k$, vertex is $(h,k)$.

Flashcard 30: State the zero-product property.

Answer: If $ab = 0$, then $a = 0$ or $b = 0$. If a product equals zero, at least one factor must be zero.

Flashcard 31: State the condition for a perfect square trinomial.

Answer: $b^2 = 4ac$. When discriminant equals zero, the quadratic is a perfect square.

Flashcard 32: Find the x-intercepts of $y = x^2 - 4$.

Answer: $x = -2, x = 2$. Set $y = 0$: $x^2 - 4 = 0$, so $x^2 = 4$.

Flashcard 33: State the zero-product property.

Answer: If $ab = 0$, then $a = 0$ or $b = 0$. If a product equals zero, at least one factor must be zero.

Flashcard 34: Convert $y = x^2 - 4x + 4$ to vertex form.

Answer: $y = (x-2)^2$. This is $(x-2)^2$ expanded, so vertex form shows vertex $(2,0)$.

Flashcard 35: What is the discriminant of $ax^2 + bx + c = 0$?

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

Flashcard 36: Write $x^2 + 8x + 16$ as a square.

Answer: $(x + 4)^2$. Perfect square trinomial with $a = x$, $b = 4$.

Flashcard 37: Determine the parabola direction for $y = 5x^2 - 3x + 2$.

Answer: Opens upwards. Positive coefficient of $x^2$ means parabola opens upward.

Flashcard 38: Find the roots of $x^2 - 5x + 6 = 0$.

Answer: $x = 2, x = 3$. Factor as $(x-2)(x-3) = 0$ to find roots.

Flashcard 39: What is the axis of symmetry for $y = ax^2 + bx + c$?

Answer: $x = \frac{-b}{2a}$. The $x$-coordinate of the vertex, found by calculus or completing the square.

Flashcard 40: Identify the $y$-intercept of $y = 2x^2 + 3x + 1$.

Answer: Intercept: $(0, 1)$. Set $x = 0$ to find where the parabola crosses the $y$-axis.

Flashcard 41: Find the discriminant for $2x^2 - 4x + 2 = 0$.

Answer: Discriminant: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

Flashcard 42: What does a positive discriminant indicate about roots?

Answer: Two distinct real roots. When $b^2 - 4ac > 0$, parabola crosses x-axis twice.

Flashcard 43: Solve $3x^2 - 12 = 0$.

Answer: $x = 2, x = -2$. Divide by 3: $x^2 - 4 = 0$, so $x = \pm 2$.

Flashcard 44: Convert $y = x^2 + 6x + 8$ to vertex form.

Answer: $y = (x+3)^2 - 1$. Complete the square: $(x+3)^2 = x^2 + 6x + 9$, so subtract 1.

Flashcard 45: Find the axis of symmetry for $y = 3x^2 + 6x + 1$.

Answer: $x = -1$. Use $x = -\frac{b}{2a} = -\frac{6}{2(3)} = -1$.

Flashcard 46: What is the effect of increasing $a$ in $y = ax^2$ on the parabola?

Answer: Narrower parabola. Larger $|a|$ values compress the parabola horizontally.

Flashcard 47: What is the discriminant in the quadratic formula?

Answer: $b^2 - 4ac$. Expression under the square root that determines root types.

Flashcard 48: What is the axis of symmetry for $y = ax^2 + bx + c$?

Answer: $x = \frac{-b}{2a}$. The $x$-coordinate of the vertex, found by calculus or completing the square.

Flashcard 49: What is the vertex of $y = 2(x-1)^2 + 3$?

Answer: Vertex: $(1, 3)$. In vertex form $y = a(x-h)^2 + k$, vertex is $(h,k)$.

Flashcard 50: What is the vertex form of a quadratic equation?

Answer: $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical shift directly.

Flashcard 51: Which term in $ax^2 + bx + c$ affects the parabola's direction?

Answer: The coefficient $a$. Positive $a$ opens up, negative $a$ opens down.

Flashcard 52: What is the standard form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. The general form where $a \neq 0$ defines a parabola.

Flashcard 53: What is the product of the roots of $ax^2 + bx + c = 0$?

Answer: $\frac{c}{a}$. By Vieta's formulas relating coefficients to roots.

Flashcard 54: Identify the $y$-intercept of $y = 2x^2 + 3x + 1$.

Answer: Intercept: $(0, 1)$. Set $x = 0$ to find where the parabola crosses the $y$-axis.

Flashcard 55: State the quadratic formula.

Answer: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Solves any quadratic by substituting coefficients $a$, $b$, and $c$.

Flashcard 56: Find the vertex of $y = x^2 - 4x + 3$.

Answer: Vertex: $(2, -1)$. Use $x = \frac{4}{2} = 2$, then $y = 4 - 8 + 3 = -1$.

Flashcard 57: Find the discriminant for $2x^2 - 4x + 2 = 0$.

Answer: Discriminant: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

Flashcard 58: Which method can solve any quadratic equation?

Answer: The quadratic formula. Works for any quadratic, unlike factoring or square roots.

Flashcard 59: Find the vertex of $y = x^2 - 4x + 3$.

Answer: Vertex: $(2, -1)$. Use $x = \frac{4}{2} = 2$, then $y = 4 - 8 + 3 = -1$.

Flashcard 60: What is the discriminant in the quadratic formula?

Answer: $b^2 - 4ac$. Expression under the square root that determines root types.

Flashcard 61: Which term in $ax^2 + bx + c$ affects the parabola's direction?

Answer: The coefficient $a$. Positive $a$ opens up, negative $a$ opens down.

Flashcard 62: Simplify: $(x + 2)^2$.

Answer: $x^2 + 4x + 4$. Apply the perfect square formula $(a+b)^2 = a^2 + 2ab + b^2$.

Flashcard 63: Identify the vertex form of a quadratic equation.

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

Flashcard 64: Which method can solve any quadratic equation?

Answer: The quadratic formula. Works for any quadratic, unlike factoring or square roots.

Flashcard 65: Find the x-intercepts of $y = x^2 - 4$.

Answer: $x = -2, x = 2$. Set $y = 0$: $x^2 - 4 = 0$, so $x^2 = 4$.

Flashcard 66: Identify the vertex form of a quadratic equation.

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

Flashcard 67: Find the roots of $x^2 - 5x + 6 = 0$.

Answer: $x = 2, x = 3$. Factor as $(x-2)(x-3) = 0$ or use the quadratic formula.

Flashcard 68: State the condition for a perfect square trinomial.

Answer: $b^2 = 4ac$. When discriminant equals zero, the quadratic is a perfect square.

Flashcard 69: What is the sum of the roots of $ax^2 + bx + c = 0$?

Answer: $-\frac{b}{a}$. By Vieta's formulas relating coefficients to roots.

Flashcard 70: Find the axis of symmetry for $y = 3x^2 + 6x + 1$.

Answer: $x = -1$. Use $x = -\frac{b}{2a} = -\frac{6}{2(3)} = -1$.

Flashcard 71: Convert $y = 2(x-3)^2 + 4$ to standard form.

Answer: $y = 2x^2 - 12x + 22$. Expand $(x-3)^2 = x^2 - 6x + 9$, then distribute and combine.

Flashcard 72: What does a positive discriminant indicate about roots?

Answer: Two distinct real roots. When $b^2 - 4ac > 0$, parabola crosses x-axis twice.

Flashcard 73: Solve $3x^2 - 12 = 0$.

Answer: $x = 2, x = -2$. Divide by 3: $x^2 - 4 = 0$, so $x = \pm 2$.

Flashcard 74: Find the roots of $x^2 - 5x + 6 = 0$.

Answer: $x = 2, x = 3$. Factor as $(x-2)(x-3) = 0$ to find roots.