This course is designed to take you from the fundamentals of SAT Math to its most advanced concepts, building confidence and problem-solving skills along the way. You’ll start with essential algebra, data analysis, and geometry, gradually progressing to complex functions, trigonometry, and high-level reasoning. With clear explanations, practical examples, and expert strategies, you'll develop the accuracy and speed needed to tackle any SAT Math challenge. Whether you're just beginning or aiming for top scores, this course will help you maximize your potential and achieve success.  

In this section, you will dive into the world of basic algebra for SAT Math, starting with simple examples and gradually building confidence. Learn how to organize your preparation in the first section: About the Digital SAT.

In this section, you will discover the keys to solving problems and analyzing data for SAT Math, beginning with fundamental ideas and progressing further.

In this section, you will take your first steps into advanced math for SAT Math, breaking down essential concepts through accessible examples.

In this section, you will get acquainted with the fundamentals of geometry and trigonometry for SAT Math, where simple shapes and angles become your allies.

In this section, you will elevate your algebra skills to an intermediate level for SAT Math, immersing yourself in interesting and diverse problems.

In this section, you will refine your ability to analyze data and solve medium-difficulty problems for SAT Math, exploring new approaches.

In this section, you will uncover the secrets of intermediate-level advanced math for SAT Math, working with unconventional ideas.

In this section, you will master intermediate-level geometry and trigonometry for SAT Math, combining logic with spatial reasoning.

In this section, you will take on the most challenging algebraic puzzles for SAT Math, where every step is a new milestone.

In this section, you will delve into the highest level of data analysis for SAT Math, unraveling complex patterns.

In this section, you will conquer the peaks of advanced math for SAT Math, exploring its most intricate aspects.

In this section, you will master the highest levels of geometry and trigonometry for SAT Math, where precision and creativity go hand in hand.

Let the given matrix be

$$
\begin{bmatrix}
4p & 2 \\
-5 & 3
\end{bmatrix}.
$$

Find its determinant.

Given the complex number  
$$
z = 5 - 3i,
$$  
where $$ i $$ is the imaginary unit. Find the value of $$ z^2 $$.

What is the solution to the equation?
$$3x - 5 = 16$$

To solve the inequality:

1. Subtract 3 from both sides:
   $$
   2a \geq 6
   $$
2. Divide both sides by 2:
   $$
   a \geq 3
   $$

The minimum value of $$ a $$ that satisfies the inequality is 3.

Lines $$ c $$ and $$ d $$ are graphed in the $$ xy $$-plane.

Line $$ c $$ is given by the equation:

$$
y = -4x + 2
$$

If line $$ d $$ is parallel to line $$ c $$ and passes through the point $$ (0, -5) $$, what is the equation of line $$ d $$?

Olga went to a store and bought a sweater originally priced at 1200 USD. The store had a promotion:

- First, a 15% discount was applied to the sweater.
- Then, an additional 10% discount was applied to the remaining amount.

What is the final price of the sweater after all the discounts?

In a library, books are divided into two genres: historical novels and popular science books in a ratio of 3:2. This means that for every 3 historical books, there are 2 popular science books.

If the library contains only these two types of books, which of the following numbers could be the total number of books in the library?

1. Calculate the total milk production of all 15 cows:  

   $$ 28 \times 15 = 420 \text{ liters} $$
   

2. Calculate the total milk production of the 5 sold cows:  
   $$
   40 \times 5 = 200 \text{ liters}
   $$

3. Find the remaining milk production after selling the cows:  
   $$
   420 - 200 = 220 \text{ liters}
   $$

4. Calculate the new average milk production per cow for the remaining 10 cows:  
   $$
   \frac{220}{10} = 22
   $$

$$ f(n) = 400 - 20n + k $$

In the function above, k is a constant. If
f(5) = 310, what is the value of f(7)?

If $$ a^4 = 50 $$ and $$ b^8 = 25 $$, what is $$ a^8 b^{-8} $$?

**Step 1: Compute the Discriminant**  
The discriminant formula:

$$
D = b^2 - 4ac
$$

Substituting the values $$ a = 1 $$, $$ b = -7 $$, $$ c = 12 $$:

$$
D = (-7)^2 - 4(1)(12) = 49 - 48 = 1
$$

**Step 2: Compute the Roots Using the Quadratic Formula**  
The quadratic formula:

$$
x = \frac{-b \pm \sqrt{D}}{2a}
$$

Substituting the values:

$$
x = \frac{-(-7) \pm \sqrt{1}}{2(1)}
$$

$$
x = \frac{7 \pm 1}{2}
$$

**Step 3: Compute the Roots**

1. **First root:**

$$
x_1 = \frac{7 + 1}{2} = \frac{8}{2} = 4
$$

2. **Second root:**

$$
x_2 = \frac{7 - 1}{2} = \frac{6}{2} = 3
$$

Thus, the roots are **$$ x_1 = 4 $$ and $$ x_2 = 3 $$**.

The sum of all values of $$ x $$ that satisfy the equation is: $$\mathbf{7}$$

A baker is making chocolate truffles, which are shaped like perfect spheres. One truffle has a radius of 6 centimeters. Find its volume in $$ \text{cm}^3 $$.

The distance between two points in space A (1, y, 2) and B (2, 1, 4) is 3. Find the possible values of y.

In a right triangle, the legs are 9 cm and 12 cm.  
Find the sine of the acute angle opposite the 9 cm leg.

$$
\begin{cases}
3x + 2y = 12 \\
5x - y = 7
\end{cases}
$$

Which ordered pair (x, y) satisfies the system of equations above?

If $$ |2x - 7| - 3 = 8 $$, what is the minimum value of x?

**Step 1: Expand the Brackets**

Using the formula $$(a \pm b)^2 = a^2 \pm 2ab + b^2$$:

- **Left side:**
  $$
  (x - 5)^2 = x^2 - 10x + 25.
  $$

- **Right side:**
  $$
  (x + 3)^2 = x^2 + 6x + 9.
  $$

**Step 2: Simplify the Equation**

Write the equation after expanding the brackets:

$$
x^2 - 10x + 25 = x^2 + 6x + 9.
$$

Subtract $$x^2$$ from both sides:

$$
-10x + 25 = 6x + 9.
$$

**Step 3: Solve the Linear Equation**

1. **Add $$10x$$ to both sides** to collect all $$x$$ terms on one side:
   $$
   25 = 16x + 9.
   $$

2. **Subtract 9 from both sides:**
   $$
   25 - 9 = 16x \quad \Rightarrow \quad 16 = 16x.
   $$

3. **Divide both sides by 16:**
$$
x = \frac{16}{16} = 1.
$$

1. It is given that **30 hectares** are cultivated by **4 tractors**. This means the ratio of field area to the number of tractors is **30:4**.  
   We need to determine how many tractors are required for **360 hectares**.

2. Set up a proportion:
   $$
   \frac{\text{hectares}}{\text{tractors}} = \frac{\text{hectares}}{\text{tractors}}
   $$
   $$
   \frac{360}{x} = \frac{30}{4}
   $$
   Where **x** represents the number of tractors.

3. Solve for **x** using cross multiplication:
   $$
   360 \times 4 = 30 \times x
   $$
   $$
   1440 = 30x
   $$

4. Divide both sides by **30** to isolate **x**:
   $$
   x = \frac{1440}{30}
   $$

5. Calculate the result:
   $$
   x = 48
   $$

When two dice are thrown, what is the probability of getting two 5s?

In 2018, the average score of students in a standardized test was 65 points, with a standard deviation of 12 points.  
In 2023, the average score of students in the same test was 72 points, with a standard deviation of 8 points.

Find the sum of the expressions:

$$
\frac{x^2}{x+3} + \frac{4x+3}{x+3}
$$

$$
\frac{a^2 - 4b^2}{a - 2b}
$$

Which of the following is equivalent to the expression above for all values of $$x$$ and $$y$$ such that $$a \neq 2b$$?

**Step 1: Write the equation for the geometric progression**  
The colony grows according to the formula of a **geometric progression**:

$$
a_n = a_1 \cdot r^{n-1}
$$

where:

- $$ a_n $$ is the number of bacteria on day $$ n $$,
- $$ a_1 $$ is the initial number of bacteria (on day 1),
- $$ r $$ is the growth factor (common ratio),
- $$ n $$ is the day number.

Given values:

- On **day 2**: $$ a_2 = 10 $$, so

  $$
  a_1 \cdot r = 10
  $$

- On **day 5**: $$ a_5 = 80 $$, so

  $$
  a_1 \cdot r^4 = 80
  $$

  **Step 2: Determine the common ratio $$ r $$**  
  Divide the second equation by the first equation:

$$
\frac{a_1 \cdot r^4}{a_1 \cdot r} = \frac{80}{10}
$$

$$
r^3 = 8
$$

$$
r = \sqrt[3]{8} = 2
$$

**Step 3: Determine the initial number of bacteria $$ a_1 $$**  
Substituting $$ r = 2 $$ into the equation $$ a_1 \cdot r = 10 $$:

$$
a_1 \cdot 2 = 10
$$

$$
a_1 = 5
$$

Archaeologists discovered an ancient amulet in the shape of a regular polygon with _n_ sides. After measurements, they found that the average interior angle of this polygon is 165°. How many sides does this polygon have?

Two angles are given: the first is $$45^\circ$$, and the second is $$135^\circ$$. Convert each angle from degrees to radians, then find the sum of the resulting values in radians.

The equation

$$
x^2 + 6x + y^2 - 4y = 3
$$

defines a circle in the $$xy$$-plane. What are the coordinates of the center of this circle?

Given the parametric equations:

$$
\begin{aligned}
x &= t^2 + t,\\
y &= t^2 - 2t.
\end{aligned}
$$

Determine which of the following statements correctly describes the possible range of values for $$ y $$ on the graph of these equations.

In the $$ xy $$-plane, a line passes through the points $$ (6,4) $$ and $$ (p,q) $$ and also passes through the origin $$ (0,0) $$. If $$ p \neq 0 $$, find the value of the expression $$ \frac{q}{p} $$.

We use the fundamental logarithm-to-exponent conversion formula:

$$
\log_b z = c \quad \Rightarrow \quad z = b^c
$$

1. Find the value of $$ x $$
$$
\log_2 x = 5
$$

$$
x = 2^5 = 32
$$

2. Find the value of $$ y $$  
$$
\log_4 y = 3
$$

$$
y = 4^3 = 64
$$

3. Compute the sum $$ x + y $$  
$$
x + y = 32 + 64 = 96
$$

At a bridge construction site, the ratio of workers to engineers is 36 to 5.

Additionally, the ratio of engineers to managers is 10 to 3.

What is the ratio of managers to workers?

A truck consumes fuel in direct proportion to the distance it travels. If the truck uses $$ f $$ liters of fuel for 120 km, how many liters of fuel will it consume for 180 km?

Imagine you have a jar containing 60 candies:

- 15 of them are chocolate candies,
- 6 of them are mint candies,
- and there is 1 candy that is both chocolate and mint.

What is the probability that a candy randomly chosen from the jar will be either chocolate or mint?

**Vector Difference Formula:**  
 For two vectors $$\vec{x} = (x_1, x_2)$$ and $$\vec{y} = (y_1, y_2)$$, the difference is given by:  
 $$
\vec{x} - \vec{y} = (x_1 - y_1,\, x_2 - y_2)
$$

**Magnitude (Norm) Formula:**  
 The magnitude of a vector $$\vec{v} = (v_1, v_2)$$ is computed as:  
 $$
\|\vec{v}\| = \sqrt{v_1^2 + v_2^2}
$$

1. **Compute the Vector Difference:**  
   $$
   \vec{x} - \vec{y} = (4 - 1,\, -1 - (-5)) = (3,\, 4)
   $$

2. **Calculate the Magnitude:**  
 $$
\|\vec{x} - \vec{y}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
$$

A sphere with a diameter of 10 meters is inscribed in a cube. Find the volume of the space between the sphere and the cube.

In the $$ xy $$-coordinate system, line $$ p $$ has a slope of $$ \dfrac{3}{4} $$ and passes through the point $$ (2,1) $$. Which of the following points **cannot** lie on line $$ p $$?

The formula for the volume of a cone is:

$$
V = \frac{1}{3} \pi r^2 h
$$

Substituting the given values:

$$
27\pi = \frac{1}{3} \pi r^2 (9)
$$

$$
27\pi = 3\pi r^2
$$

Dividing both sides by $$ 3\pi $$:

$$
r^2 = \frac{27\pi}{3\pi} = 9
$$

Taking the square root:

$$
r = \sqrt{9} = 3
$$

Quiz: Foundations: Algebra

Explanation