Q: How do you factor a quadratic such as $ x^{2} - 5x + 6 $?

Find two numbers that multiply to the constant term and add to the middle coefficient. For $ x^{2} - 5x + 6 $, the numbers $ -2 $ and $ -3 $ multiply to $ 6 $ and add to $ -5 $, so it factors as $ (x - 2)(x - 3) $. Setting each factor to zero gives the roots $ x = 2 $ and $ x = 3 $. Writing expressions in equivalent forms by factoring is a Principles of Algebra topic on the 5162. Recognize special forms: a difference of squares factors as $ a^{2} - b^{2} = (a + b)(a - b) $, and always pull out the greatest common factor first.

Q: When should you use the quadratic formula instead of factoring?

Use the quadratic formula $ x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a} $ when a quadratic $ ax^{2} + bx + c = 0 $ does not factor over the integers. The discriminant $ b^{2} - 4ac $ reveals the roots: positive gives two real roots, zero gives one repeated root, and negative gives no real roots. Solving quadratics by multiple techniques — factoring, completing the square, and the quadratic formula — is a Principles of Algebra skill on the 5162. Completing the square rewrites $ ax^{2} + bx + c = 0 $ in the form $ (x - p)^{2} = q $, which both solves the equation and reveals the vertex.

Question 1

How many questions are on the Praxis Algebra I (5162) test?

Accepted Answer

The Praxis Algebra I (5162) has 60 selected-response questions, including both scored and unscored pretest items. Some questions ask you to select one answer and others to select one or more. There is no penalty for wrong answers, so answer every question.

Question 2

How long is the Praxis 5162 test and is a calculator provided?

Accepted Answer

You have 150 minutes of testing time for the 60 questions, which is 2.5 minutes per question. The 5162 is computer-delivered and provides an on-screen graphing calculator — you may not bring your own handheld calculator.

Question 3

What content does the Praxis 5162 cover?

Accepted Answer

Three ETS content categories: Principles of Algebra (about 23 questions, 38%); Functions (about 18 questions, 30%); and Number and Quantity, Probability and Statistics (about 19 questions, 32%). The math is secondary Algebra I content, aligned to the NCTM standards.

Question 4

How is the Praxis 5162 scored and what is a passing score?

Accepted Answer

Raw correct answers convert to a scaled score from 100 to 200. There is no single national passing score — each state or licensing agency sets its own cut score, commonly in the 150s. Always confirm the requirement for the state where you plan to teach.

Question 5

Who needs to take the Praxis Algebra I (5162)?

Accepted Answer

The 5162 is for prospective secondary mathematics teachers seeking certification to teach Algebra I. It assesses the algebra, functions, and number/statistics knowledge an entry-level Algebra I teacher must have, aligned to the NCTM CAEP and PSSM standards.

Question 6

How should I use this Praxis 5162 study guide?

Accepted Answer

Work through the three content categories in order — Principles of Algebra, Functions, then Number, Quantity & Statistics. After each module take the checkpoint quiz to find gaps, then drill that area with our free practice questions and flashcards, and revisit flagged sections before test day.

Question 7

Is this Praxis 5162 study guide free?

Accepted Answer

Yes — the full guide, the checkpoints, the glossary, the practice questions, and the flashcards are 100% free, with no account required.

Question 8

How do you solve a two-step linear equation like $ 3x - 7 = 11 $?

Accepted Answer

Undo the operations in reverse order, keeping both sides balanced. Add 7 to both sides to get $ 3x = 18 $, then divide both sides by 3 to get $ x = 6 $. Check by substituting: $ 3(6) - 7 = 11 $. The same balancing logic extends to multi-step and variables-on-both-sides equations. Justifying each step in solving an equation is a Principles of Algebra skill on the 5162. An inequality solves the same way, but multiplying or dividing both sides by a negative number flips the inequality sign — for example $ -2x > 6 $ gives $ x < -3 $.

Question 9

How do you factor a quadratic such as $ x^{2} - 5x + 6 $?

Accepted Answer

Find two numbers that multiply to the constant term and add to the middle coefficient. For $ x^{2} - 5x + 6 $, the numbers $ -2 $ and $ -3 $ multiply to $ 6 $ and add to $ -5 $, so it factors as $ (x - 2)(x - 3) $. Setting each factor to zero gives the roots $ x = 2 $ and $ x = 3 $. Writing expressions in equivalent forms by factoring is a Principles of Algebra topic on the 5162. Recognize special forms: a difference of squares factors as $ a^{2} - b^{2} = (a + b)(a - b) $, and always pull out the greatest common factor first.

Question 10

When should you use the quadratic formula instead of factoring?

Accepted Answer

Use the quadratic formula $ x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a} $ when a quadratic $ ax^{2} + bx + c = 0 $ does not factor over the integers. The discriminant $ b^{2} - 4ac $ reveals the roots: positive gives two real roots, zero gives one repeated root, and negative gives no real roots. Solving quadratics by multiple techniques — factoring, completing the square, and the quadratic formula — is a Principles of Algebra skill on the 5162. Completing the square rewrites $ ax^{2} + bx + c = 0 $ in the form $ (x - p)^{2} = q $, which both solves the equation and reveals the vertex.

Question 11

How do you solve a system of two linear equations?

Accepted Answer

Use substitution or elimination, or read the intersection point of the two graphs. For $ y = 2x + 1 $ and $ y = -x + 4 $, set them equal: $ 2x + 1 = -x + 4 $, so $ 3x = 3 $ and $ x = 1 $, giving $ y = 3 $. The solution $ (1, 3) $ is where the two lines cross. Solving systems algebraically and graphically is a Principles of Algebra topic on the 5162. The intersection of the graphs of $ y = f(x) $ and $ y = g(x) $ is the solution of $ f(x) = g(x) $ — this includes linear-quadratic systems.

Question 12

How do you rearrange a formula to solve for a different variable?

Accepted Answer

Treat the target variable like an unknown and isolate it with inverse operations, exactly as you would solve an equation. To solve $ d = rt $ for $ t $, divide both sides by $ r $: $ t = \dfrac{d}{r} $. This literal-equation skill lets you create equations that model constraints and relationships. Creating equations and rearranging formulas to highlight a quantity of interest is a Principles of Algebra skill on the 5162. The same moves graph linear inequalities: solve for $ y $, graph the boundary line, then shade the half-plane that satisfies the inequality.

Question 13

What do domain and range mean, and how do you evaluate a function?

Accepted Answer

The domain is the set of allowed inputs and the range is the set of resulting outputs. To evaluate a function, substitute the input for the variable: for $ f(x) = 3x^{2} - 2x + 1 $, $ f(-1) = 3(-1)^{2} - 2(-1) + 1 = 3 + 2 + 1 = 6 $. Function notation $ f(x) $ names the output for a given input $ x $. Understanding the function concept and notation is a Functions skill on the 5162. Restrict the domain where an expression breaks down — for example a rational function excludes inputs that make the denominator zero, and a square-root function excludes negatives.

Question 14

What do the slope and intercepts of a linear function tell you?

Accepted Answer

In slope-intercept form $ y = mx + b $, the slope $ m = \dfrac{\Delta y}{\Delta x} $ is the constant rate of change and $ b $ is the $ y $-intercept, the output when $ x = 0 $. The $ x $-intercept is where $ y = 0 $. For $ y = 2x - 1 $, the line rises 2 for every 1 across and crosses the $ y $-axis at $ -1 $. Interpreting linear functions and their slope and intercepts is a Functions topic on the 5162. The average rate of change of a function on $ [a, b] $ is $ \dfrac{f(b) - f(a)}{b - a} $; for a line it is constant, but for a curve it varies.

Question 15

What are the key features of a quadratic function's graph?

Accepted Answer

The graph of $ f(x) = ax^{2} + bx + c $ is a parabola. Its axis of symmetry is the vertical line $ x = -\dfrac{b}{2a} $, and the vertex sits on that line at the minimum (if $ a > 0 $) or maximum (if $ a < 0 $). The roots, or zeros, are where the parabola crosses the $ x $-axis. Analyzing quadratic functions — zeros, extrema, and symmetry — is a Functions skill on the 5162. Vertex form $ f(x) = a(x - h)^{2} + k $ reveals the vertex $ (h, k) $ directly, while factored form reveals the zeros.

Question 16

How do transformations change the graph of a function?

Accepted Answer

Starting from $ f(x) $: $ f(x) + k $ shifts the graph up by $ k $; $ f(x + k) $ shifts it left by $ k $; $ k \cdot f(x) $ stretches it vertically; and $ f(kx) $ compresses it horizontally. A negative factor reflects the graph across an axis. Recognizing these lets you build new functions from a known parent graph. Building new functions from existing ones through transformations is a Functions topic on the 5162. A function is even when $ f(-x) = f(x) $ (symmetric about the $ y $-axis) and odd when $ f(-x) = -f(x) $ (symmetric about the origin).

Question 17

How do linear, quadratic, and exponential models differ?

Accepted Answer

A linear function grows by equal differences over equal intervals (constant slope), while an exponential function grows by equal factors (constant ratio), as in $ A(t) = Pe^{rt} $. A quadratic grows by a changing rate whose differences are themselves constant. Over time, exponential growth always overtakes linear and quadratic growth. Distinguishing linear, quadratic, and exponential models is a Functions skill on the 5162. An arithmetic sequence adds a constant difference each term; a geometric sequence multiplies by a constant ratio. Sequences are functions whose domain is the integers.

Question 18

How do rational exponents relate to radicals?

Accepted Answer

A rational exponent is another way to write a radical: $ a^{1/n} = \sqrt[n]{a} $ and $ a^{m/n} = \sqrt[n]{a^{m}} $. For example $ 8^{2/3} = \left(\sqrt[3]{8}\right)^{2} = 2^{2} = 4 $. A negative exponent means a reciprocal, $ a^{-n} = \dfrac{1}{a^{n}} $, and the exponent laws apply to both forms. Working with the properties of radicals and exponents is a Number and Quantity skill on the 5162. Use scientific notation $ a \times 10^{n} $ and order of magnitude to handle very large or very small numbers efficiently.

Question 19

What is the difference between rational and irrational numbers?

Accepted Answer

A rational number can be written as a ratio of two integers $ \dfrac{a}{b} $ with $ b \neq 0 $; its decimal terminates or repeats. An irrational number, such as $ \sqrt{2} $ or $ \pi $, cannot. Closure rules follow: rational $ \pm $ rational is rational, but a nonzero rational times an irrational is always irrational. Reasoning about the properties of rational and irrational numbers is a Number and Quantity topic on the 5162. The sum of a rational and an irrational number is always irrational — a common closure question on the exam.

Question 20

How do you find the mean, median, and range of a data set?

Accepted Answer

The mean is the average, $ \bar{x} = \dfrac{\sum x}{n} $; the median is the middle value of the ordered data; and the range is the largest value minus the smallest. For $ 4, 7, 5, 8, 6 $ the mean is $ \dfrac{30}{5} = 6 $, the median is 6, and the range is $ 8 - 4 = 4 $. The median resists outliers; the mean does not. Interpreting measures of center and spread is a Probability and Statistics topic on the 5162. Data displays — dot plots, histograms, box plots, and scatter plots — summarize a distribution's shape, center, and spread visually.

Question 21

How do you compute the probability of a simple or compound event?

Accepted Answer

Probability is favorable outcomes over total equally likely outcomes, a value from 0 to 1. Rolling a 3 on a fair die has probability $ \dfrac{1}{6} $. For independent events, multiply: two heads on two flips is $ \dfrac{1}{2} \cdot \dfrac{1}{2} = \dfrac{1}{4} $. For mutually exclusive events, add the probabilities. Simple and compound probability is a Probability and Statistics topic named in the third 5162 content category. Reason quantitatively and attend to units: keep units consistent in formulas and check that an answer's scale is reasonable.

Detail	Praxis Algebra I (5162)
Questions	60 selected-response (select one or more answers; includes unscored pretest items)
Time	150 minutes of testing time
Content	Principles of Algebra (~23, 38%), Functions (~18, 30%), Number & Quantity; Probability & Statistics (~19, 32%)
Score scale	100–200 scaled; passing score set by each state (commonly ~150s)
Calculator	On-screen graphing calculator provided; own calculator not allowed
Guessing penalty	None — answer every question
Delivery	Computer-delivered, at a test center or online with proctoring
Test fee	$130 (verify current fee at ets.org — fees change)
Publisher	ETS (Educational Testing Service)

Idea	Rule
Slope / rate of change	$m = \dfrac{\Delta y}{\Delta x}$
Average rate of change	$\dfrac{f(b) - f(a)}{b - a}$
Vertex form of a parabola	$a(x - h)^{2} + k,\ \text{vertex } (h, k)$
Axis of symmetry	$x = -\dfrac{b}{2a}$
Exponential model	$y = a \cdot b^{x},\quad A(t) = Pe^{rt}$

Evaluating the rule
x	f(x)
−1	6
0	1
1	2
2	9

Praxis 5162 Exam Snapshot

1 · Principles of Algebra

Expressions, Exponents & Operations

Linear Equations & Inequalities

Systems of Equations

Polynomials & Factoring

Quadratics & Radicals

Checkpoint · Category · Principles of Algebra

2 · Functions

Function Concept & Notation

Linear Functions

Quadratic & Other Functions

Interpreting Graphs & Sequences

Checkpoint · Category · Functions

3 · Number, Quantity, Probability & Statistics

The Real Number System & Quantity

Ratios & Proportional Reasoning

Statistics — Center, Spread & Displays

Probability

Checkpoint · Category · Number, Quantity & Statistics

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