SAT Math — Quadratic Equations and Functions

What Is a Quadratic?

A quadratic is any equation or function where the highest power of the variable is 2. The standard form is:

y = ax² + bx + c

Quadratics produce parabolas (U-shaped curves) when graphed. They appear in roughly 10–15% of SAT math questions.

Three Key Forms of a Quadratic

| Form | Equation | What it tells you directly | |---|---|---| | Standard | y = ax² + bx + c | Y-intercept (c); direction (a > 0 = opens up, a < 0 = opens down) | | Factored | y = a(x − r)(x − s) | Roots/x-intercepts: r and s | | Vertex | y = a(x − h)² + k | Vertex (h, k); axis of symmetry x = h |

You need to be comfortable converting between all three forms.

Finding Roots: Factoring

When ax² + bx + c = 0, find two numbers that:

• Multiply to give a × c
• Add to give b

Example: x² + 5x + 6 = 0

• Multiply to 6, add to 5: the numbers are 2 and 3
• Factor: (x + 2)(x + 3) = 0
• Solutions: x = −2 or x = −3

The Quadratic Formula

When factoring doesn't work cleanly, use the quadratic formula:

x = (−b ± √(b² − 4ac)) / (2a)

The part under the radical, b² − 4ac, is called the discriminant:

• Positive discriminant: Two real solutions (two x-intercepts)
• Zero discriminant: One real solution (parabola touches x-axis at exactly one point — the vertex)
• Negative discriminant: No real solutions (parabola doesn't cross x-axis)

The Vertex

The vertex is the highest or lowest point of the parabola.

From vertex form: y = a(x − h)² + k → vertex is (h, k) From standard form: x-coordinate of vertex = −b / (2a)

Example: y = 2x² − 8x + 3 > x-coordinate of vertex: −(−8) / (2×2) = 8/4 = 2 > y-coordinate: y = 2(2)² − 8(2) + 3 = 8 − 16 + 3 = −5 > Vertex: (2, −5)

Real-world example: A ball is thrown upward and its height in feet is modeled by h(t) = −16t² + 64t + 5, where t is seconds. The maximum height is at the vertex. x-coordinate: −64/(2×−16) = −64/−32 = 2 seconds. Height at t = 2: h(2) = −16(4) + 64(2) + 5 = −64 + 128 + 5 = 69 feet.

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Key Terms

• Quadratic: An equation with x²; produces a parabola when graphed
• Standard form: ax² + bx + c
• Factored form: a(x − r)(x − s) where r and s are the roots
• Vertex form: a(x − h)² + k where (h, k) is the vertex
• Roots / zeros / x-intercepts: The values of x where the parabola crosses the x-axis
• Vertex: The turning point of the parabola; maximum if a < 0, minimum if a > 0
• Discriminant: b² − 4ac; determines number of real solutions
• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)
• Axis of symmetry: The vertical line through the vertex: x = −b/(2a)

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Quiz Questions:

Q1. Solve by factoring: x² − 7x + 12 = 0

A) x = 3 or x = 4 B) x = −3 or x = −4 C) x = −3 or x = 4 D) x = 3 or x = −4

Answer: A — Find numbers that multiply to 12 and add to −7: −3 and −4. Factor: (x − 3)(x − 4) = 0. Solutions: x = 3 or x = 4.

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Q2. The equation y = (x − 5)(x + 1) is in factored form. What are the x-intercepts?

A) x = 5 and x = −1 B) x = −5 and x = 1 C) x = 5 and x = 1 D) x = −5 and x = −1

Answer: A — Set each factor equal to zero: x − 5 = 0 → x = 5; x + 1 = 0 → x = −1. X-intercepts are (5, 0) and (−1, 0).

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Q3. What is the vertex of y = 3(x − 2)² + 7?

A) (2, 7) B) (−2, 7) C) (2, −7) D) (3, 7)

Answer: A — This is vertex form y = a(x − h)² + k. Here h = 2, k = 7. Vertex = (h, k) = (2, 7).

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Q4. The discriminant of a quadratic equation is calculated and equals −9. What does this mean about the equation's solutions?

A) The equation has two equal real solutions B) The equation has two distinct real solutions C) The equation has no real solutions D) The equation has one real solution and one imaginary solution

Answer: C — When the discriminant (b² − 4ac) is negative, there are no real solutions (the parabola doesn't intersect the x-axis). All solutions are complex/imaginary in this case.

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Q5. A quadratic equation has the form y = −2x² + 12x − 10. What is the x-coordinate of the vertex?

A) x = 3 B) x = 6 C) x = −3 D) x = −6

Answer: A — Use x = −b/(2a) = −12/(2×−2) = −12/−4 = 3.