Algebra 1 — 20 Essential Problems

Study Guide

Core Concepts & Formulas

Evaluating Expressions

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Substitute the given value for each variable and simplify using order of operations (PEMDAS).

Step 1 · Replace the variable with the given number
Step 2 · Apply PEMDAS: Parentheses → Exponents → Multiply/Divide → Add/Subtract

⚡ PEMDASSubstitute first

Example: Evaluate \(3x^2 - 2x + 1\) at \(x = -2\).
\(3(-2)^2 - 2(-2) + 1 = 3(4) + 4 + 1 = 12 + 4 + 1 = \mathbf{17}\)

Solving Linear Equations

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Use inverse operations to isolate the variable. Whatever you do to one side, do to the other.

ax + b = c → ax = c − b → x = (c − b) / a

Balance both sidesInverse ops

Example: Solve \(2x - 5 = 11\).
\(2x = 16\) → \(x = 8\)

Distributive Property & Simplifying

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Distribute the coefficient to each term inside the parentheses, then combine like terms.

a(b + c) = ab + ac

Distribute → Combine

Example: Simplify \(3(2x-4)-2(x+1)\).
\(= 6x - 12 - 2x - 2 = \mathbf{4x - 14}\)

Slope & Linear Equations

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Slope measures the rate of change. Slope-intercept form is the most useful linear equation form.

Slope: \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)

Slope-intercept: \(y = mx + b\) (\(b\) = y-intercept)

Rise over Runy = mx + b

Example: Slope between \((1,-1)\) and \((3,7)\):
\(m = \dfrac{7-(-1)}{3-1} = \dfrac{8}{2} = \mathbf{4}\)

Inequalities

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Solve like an equation — but flip the inequality sign when multiplying or dividing by a negative number.

If \(-a \cdot x > b\), then \(x < -b/a\) (FLIP!)

Flip when ÷ by negative

Example: \(4x + 3 > 15\)
\(4x > 12\) → \(x > 3\)

Factoring Quadratics

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For \(x^2 + bx + c\), find two numbers that multiply to \(c\) and add to \(b\).

\(x^2 + bx + c = (x+p)(x+q)\) where \(p \cdot q = c\) and \(p + q = b\)

Multiply → cAdd → b

Example: Factor \(x^2 + 5x + 6\).
Find: \(2 \times 3 = 6\), \(2 + 3 = 5\) → \(\mathbf{(x+2)(x+3)}\)

Systems of Equations

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Solve two equations simultaneously using substitution or elimination.

Elimination: Add/subtract equations to cancel a variable.
Substitution: Express one variable in terms of the other.

EliminationSubstitution

Example: \(x+y=10\) and \(x-y=4\).
Add: \(2x=14\) → \(x=7\), then \(y=3\).

Functions & Evaluation

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A function assigns exactly one output to each input. Evaluate by substituting the input value.

\(f(a)\) means substitute \(x = a\) into \(f(x)\).

One input → one output

Example: \(f(x) = 2x^2 - 3x + 1\), find \(f(3)\).
\(= 2(9) - 9 + 1 = 18 - 9 + 1 = \mathbf{10}\)

Radicals & Distance Formula

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Simplify radicals by finding perfect square factors. Use the distance formula from the Pythagorean theorem.

\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)

Distance: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

Find perfect squaresPythagorean root

Example: \(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\)

Sequences, Percent & Absolute Value

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Arithmetic: \(a_n = a_1 + (n-1)d\)

Percent: \(\text{part} = \dfrac{\%}{100} \times \text{whole}\)

\(|x| = c\) means \(x = c\) or \(x = -c\)

a₁ + (n−1)d±c for absolute value

Example: Sequence 4, 7, 10, … → 10th term = \(4 + 9(3) = 31\)

Algebra 1 — 20 Essential Problems | TopEduPrep

Q 01 Evaluating Expressions ★★☆

Evaluate \(3x^2 - 2x + 1\) when \(x = -2\).

A) 9 B) 17 C) 11 D) 21

Q 02 Linear Equations ★☆☆

Solve for \(x\): \(\quad 2x - 5 = 11\)

A) 3 B) 6 C) 8 D) 10

Q 03 Distributive Property ★★☆

Simplify: \(\quad 3(2x - 4) - 2(x + 1)\)

A) 8x − 10 B) 4x − 14 C) 4x + 14 D) 6x − 14

Q 04 Slope ★★☆

What is the slope of the line passing through the points \((1, -1)\) and \((3, 7)\)?

A) 2 B) 3 C) 4 D) 8

Q 05 Linear Equations / Intercepts ★★☆

What is the \(y\)-intercept of the line \(3x - 2y = 12\)?

A) −6 B) 4 C) 6 D) −4

Q 06 Inequalities ★☆☆

Solve the inequality: \(\quad 4x + 3 > 15\)

A) x < 3 B) x > 4 C) x > 3 D) x < 4

Q 07 Factoring ★★☆

Factor completely: \(\quad x^2 + 5x + 6\)

A) (x+1)(x+6) B) (x+2)(x+3) C) (x+5)(x+1) D) (x−2)(x−3)

Q 08 Quadratic Equations ★★★

What are the solutions of \(x^2 - x - 6 = 0\)?

A) x = 2 and x = 3 B) x = 1 and x = −6 C) x = −1 and x = 6 D) x = −2 and x = 3

Q 09 Systems of Equations ★★★

Solve the system of equations:
\(x + y = 10\)
\(x - y = 4\)

A) x=7, y=3 B) x=8, y=2 C) x=6, y=4 D) x=5, y=5

Q 10 Slope-Intercept Form ★★☆

Which equation represents the line that passes through \((0, 3)\) with slope \(2\)?

A) y = 3x + 2 B) y = 2x + 3 C) y = −2x + 3 D) y = 2x − 3

Q 11 FOIL / Polynomial Multiplication ★★☆

Expand and simplify: \(\quad (x + 3)(x - 4)\)

A) x² − 7x − 12 B) x² + 7x − 12 C) x² − x − 12 D) x² + x − 12

Q 12 Absolute Value Equations ★★★

Solve: \(\quad |2x - 3| = 7\)

A) x = 2 only B) x = 5 only C) x = −5 and x = 2 D) x = 5 and x = −2

Q 13 Functions ★★☆

If \(f(x) = 2x^2 - 3x + 1\), what is the value of \(f(3)\)?

A) 8 B) 10 C) 14 D) 16

Q 14 Proportions / Ratios ★☆☆

Solve for \(x\): \(\quad \dfrac{3}{x} = \dfrac{9}{15}\)

A) 5 B) 3 C) 45 D) 6

Q 15 Percent Problems ★☆☆

What is 35% of 80?

A) 24 B) 35 C) 28 D) 32

Q 16 Arithmetic Sequences ★★☆

In an arithmetic sequence, the first term is 4 and the common difference is 3.
What is the 10th term?

A) 27 B) 28 C) 30 D) 31

Q 17 Simplifying Radicals ★★☆

Simplify: \(\quad \sqrt{72}\)

A) 8√3 B) 6√2 C) 36√2 D) 4√18

Q 18 Distance Formula ★★★

What is the distance between the points \((-1, 2)\) and \((3, -2)\)?

A) 2√8 B) 4 C) 4√2 D) 6

Q 19 Multi-Step Equations ★★☆

Solve for \(x\): \(\quad 2(x - 3) = 4x - 10\)

A) 2 B) −2 C) 4 D) −4

Q 20 Word Problems / Rate ★★☆

A car travels 240 miles in 4 hours at a constant speed.
What is the car's speed in miles per hour?

A) 45 mph B) 60 mph C) 80 mph D) 120 mph

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