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5 min read•august 25, 2021

These questions (both with-calculator and no-calculator) test on four major content areas: **the Heart of Algebra**, **Problem Solving**, **Data Analysis**, and the **Passport to Advanced Math**.

The SAT Math section contains approximately 60 questions, separated into:

- No-Calculator Section
- 15 Multiple Choice
- 5 Grid-Ins
- 25 Minutes
- Calculator Section
- 30 Multiple Choice
- 8 Grid-Ins
- 55 Minutes
**Total Time: 80 Minutes**

- Quadratic Equations & Expressions
- Function Interpretation
- Graph Interpretation
- Roots
- Square Roots
- Squaring

- Use the quadratic formula
- Factor and multiply polynomial expressions (word problems, etc.)
- Interpret graphs and functions to write quadratic expressions
- Determine the roots & square roots of various functions and equations

**Directions for the Non-Calculator Section**

- The use of a calculator
**is not permitted.** - All variables and expressions used represent real numbers unless otherwise indicated.
- Figures provided in this test are drawn to scale unless otherwise indicated.
- All figures lie in a plane unless otherwise indicated.

(a) 4(s2 − t )(4s2 + t )

(b) 4(4s2 - t)(s2 - t)

(c) 4(2s2 − t )(2s2 + t )

(d) (8s2 − 2t )(8s2 + 2t )

The correct answer is **C. **

- Recognize something familiar about this problem?
**It’s a****difference of squares!** - This classic quadratic pattern draws from x2 - y2, which can be factored into (x-y)(x+y). Recognizing these patterns can help you answer questions quickly and help you set a steady pace on the exam.
- How does this question demonstrate an understanding of the difference of squares (more specifically, the difference of perfect squares)?
- Let’s break it down!

⇒ 16s4 - 4t2

⇒ = (4s2)2 - (2t)2 (*difference of perfect squares) *

⇒ = (4s2 − 2t ) (4s2 + 2t) (*an extension of (x+y)(x-y))*

⇒ = 2(2s2 − t )(2)(2s2 + t ) (*factor out a 2!) *

⇒ = 4(2s2 − t ) (2s2 + t ) (*Multiply your factors of 2 = 4!) *

⇒ This leads us to **Option C! **

- This question tests your ability to rewrite expressions, and add/subtract/multiply polynomial expressions.

Feeling good? Let’s try another one!

(a) 2

(b) 3

(c) 7

(d) 9

The correct answer is **B. **

- This question tests your ability to
**utilize the distributive property**. Let’s expand this equation! - The key to this question is to convert one side of the equation to match the other side in order to compare like terms and determine the value of b!
**Left Hand Side =**(x2 + bx − 2)(x + 3)

⇒ = (x3 + bx2 − 2x ) + (3x2 + 3bx − 6)

⇒ = x3 + (3 + b) x2 + (3b − 2)(x − 6)

**Right Hand Side = x3 + 6x2 + 7x − 6**

⇒ Now that all like terms have been combined and expanded on the Left Hand Side, we can compare them to the Right Hand Side! It’s visible that (3+b) = 6, therefore **b = 3**

- This question tests your ability to rewrite expressions, and add/subtract/multiply polynomial expressions.

Time for Question 3! Looking good!

(a) (−1 ± √11)/2

(b) (−1 ± _√13)/2

(c) (1 ± √11)/2

(d) (1 ± √13)/2

The correct answer is **D. **

- This question draws on your ability to use the
**quadratic formula**and**completing the square.**How so? Let’s find out! - First, subtract x from each side to convert the equation into
**standard form** - Standard Form: ax2 + bc + c
- Now, you’re left with x2 - x - 3 = 0
- Comparing this to the standard form above, you can note that a = 1, b = -1, and c = -3!
- You can plug these values into the
**quadratic formula,**and simplify to get**Choice D!**

- This question tests your ability to utilize radicals, and apply the quadratic formula.

Time for Question 4! How are you feeling?

(a) 5

(b) 20

(c) −5 and 20

(d) 5 and 20

The correct answer is **B. **

- This question draws on your ability to use the
**radicals**and**use the quadratic equation.**Let’s break it down: - x − 12 = √(x + 44)
- First, square both sides of the equation!

⇒ = (x - 12)2 = (x + 44)

⇒ = x2 - 24x + 144 = x + 144

⇒ x2 - 25x + 100 = 0

⇒ (x-5)(x-20)

Seems like 5 and 20 both satisfy our requirements! But wait, there’s one more check! We need to **plug both 5 and 20 back into the original equation **to ensure they work.

After plugging both 5 and 20 in, only 20 works, therefore **Choice B is correct! **

- This question tests your ability to utilize radicals, and apply the principles of quadratic equations & factoring.

Congratulations! You’ve made it to the end of this prep activity 🙌. You learned about **“SAT Math - Quadratic Equations & Radicals”.** You should have a better understanding of the Math sections for the SAT© , topic highlights, what you will have to be able to do in order to succeed, as well as have seen some practice questions that put the concepts in action. Good luck studying for the SAT Math section 👏!

Need more resources? Check out our complete **SAT Math Study Guide w/ Practice Problems**. Pressed on time? Access our **SAT cram sessions** and watch the **Night Before the SAT cram session**. You got this 🥳.

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