Digital SAT · Math · Unit 2 · Advanced Math

Lesson 2.5 · Quadratics

On the test: the biggest Advanced Math topic — 3–5 questions, and most of the hardest ones

Big idea. A quadratic is a U-shaped curve (a parabola). It can be written three ways, and each way SHOWS something different: standard ax² + bx + c shows the y-intercept c; factored a(x − r)(x − s) shows the x-intercepts; vertex a(x − h)² + k shows the turning point. Half the trick is choosing the right form.

🎛 Move it · the parabola lab

Slide $a$ , $h$ , $k$ . Watch the vertex form, the standard form, the number of $x$ -intercepts and the min/max all change together.

y = 1 (x - 2)^{2} - 4

= 1 x^{2} - 4 x + 0

a1h2k-4

b² − 4ac = 16 → positive: TWO x-intercepts

Opens up → MINIMUM value -4 at x = 2. Sum of the roots = −b/a = 4, product = c/a = 0.

🧑‍🏫 Rule 1 · Read each formstep 1 of 5

1
📋 The question · SAT · June 2024 · Math 800 Book Q59
$y$ -intercept of $y = 2 x^{2} + 6 x + 3$ ?
1. A) $(0, 2)$
2. B) $(0, 3)$
3. C) $(0, 6)$
4. D) $(0, 9)$
standard
$y = 2 x^{2} + 6 x + 3 \Rightarrow (0, 3)$
June 2024: put $x = 0$ — everything with $x$ vanishes, $c$ is left.

🧑‍🏫 Rule 2 · Build the equation from factsstep 1 of 5

1
📋 The question · SAT · June 2023 · Math 800 Book Q6
Quadratic $g$ : vertex $(- 1, - 4)$ , through $(- 2, - 43)$ and $(1, - 160)$ . What is $g (0) - g (2)$ ?
1. A) $- 121$
2. B)0
3. C)117
4. D)312
the question
$vertex (- 1, - 4), through (1, - 160)$
June 2023.

🧑‍🏫 Rule 3 · Factor, then each factor = 0step 1 of 4

1
📋 The question · SAT · October US 2024 · Math 800 Book Q100
$x (x - 15) = 34$ . Positive solution?
(type the answer)
the question
$x (x - 15) = 34$
October US 2024. First make one side 0.

🧑‍🏫 Rule 4 · The quadratic formulastep 1 of 4

1
the formula
$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Works for every quadratic, even ones that do not factor.

🧑‍🏫 Rule 5 · How many solutions? Look at b² − 4acstep 1 of 6

1
why
$b^{2} - 4 a c$
In the formula the $\pm$ makes two answers — unless the root is 0 (one answer) or the root of a negative (no real answer).

Desmos move · count by graphing

Type

y = 4 x^{2} + k x + 9

with a slider for

k

. Slide until the parabola just TOUCHES the

x

-axis — that

k

gives one solution.

🧑‍🏫 Rule 6 · Sum = −b/a, product = c/astep 1 of 4

1
why
$a (x - r) (x - s) = a x^{2} - a (r + s) x + a r s$
Expand the factored form and match it with $a x^{2} + b x + c$ .

🧑‍🏫 Rule 7 · Treat a repeated chunk as one letterstep 1 of 5

1
📋 The question · SAT · June 2023 · Math 800 Book Q9
If $(x + 3)^{2} = 30$ , what is $x^{2} + 6 x$ ?
1. A)21
2. B)30
3. C)39
4. D)60
the question
$(x + 3)^{2} = 30, x^{2} + 6 x = ?$
June 2023.

🧑‍🏫 Rule 8 · Exactly one intersection = discriminant 0step 1 of 4

1
📋 The question · SAT · August US 2024 · Math 800 Book Q53
$y = 2 x^{2} - 19 x + 40$ and $y = x + a$ intersect at exactly one point. What is $x$ ?
1. A) $- 10$
2. B) $- 5$
3. C)5
4. D)10
the question
$y = 2 x^{2} - 19 x + 40, y = x + a$
August US 2024: they meet at exactly one point. Find $x$ .

🧑‍🏫 Rule 9 · The √(k − x) = c − x familystep 1 of 5

1
📋 The question · SAT · March 2024 · Math 800 Book Q22
In $k - x = 52 - x$ , $k$ is a constant and the equation has exactly one real solution. What is the minimum possible value of $4 k$ ?
(type the answer)
the question
$k - x = 52 - x, exactly one solution$
March 2024: minimum $4 k$ .

Real question 1SAT · December 2024 · Math 800 Book Q135

$figure from the test$

The graph of a system of a linear equation and a quadratic equation is shown. Which of the following is a solution $(x, y)$ to the system?

🧑‍🏫 Rule 10 · What each number in a height model meansstep 1 of 5

1
📋 The question · SAT · March 2024 · Math 800 Book Q33
$h = - 4.9 t^{2} + 13 t + 15$ . Which number is the height from which the object was kicked?
1. A)0
2. B)4.9
3. C)13
4. D)15
the model
$h = - 4.9 t^{2} + 13 t + 15$
March 2024: an object kicked from a platform.

🧑‍🏫 Rule · Complete the squarestep 1 of 4

1
📋 The question · Gen Z Study
The function $f (x) = x^{2} - 8 x + 21$ can be written as $f (x) = (x - h)^{2} + k$ . What is $k$ ?
(type the answer)
half the middle
$- 8 \div 2 = - 4$
Take half of the $x$ -coefficient.

🧑‍🏫 Rule · The vertex sits halfway between the zerosstep 1 of 2

1
📋 The question · Gen Z Study
What is the $x$ -coordinate of the vertex of $y = (x - 2) (x + 8)$ ?
(type the answer)
zeros
$x = 2, x = - 8$

Take with you

▸Standard shows $c$ (y-intercept); factored shows the roots; vertex form shows the turning point.
▸Vertex $x = - \frac{b}{2 a}$ ; plug it in for the min/max value.
▸Solutions: factor and set each factor to 0, or use the formula.
▸ $b^{2} - 4 a c$ : positive 2, zero 1, negative 0 solutions.
▸Sum of roots $- \frac{b}{a}$ , product $\frac{c}{a}$ .
▸Line meets parabola once → set equal, discriminant 0 (the root is the new vertex).
▸In models: the constant is the start, the vertex is the max/min.

Sign In to Keep Going