High School Algebra Practice Test

Review essential Algebra 1 and Algebra 2 skills through a balanced set of equations, functions, graphs, factoring, and applications.

High School Algebra Practice Test

20 mixed high-school algebra questions with instant explanations and a final score.

Instant feedback · Worked explanations

High School Algebra Learning Map

A balanced review should connect equations, functions, graphs, factoring, and applications.

High-school algebra mixes several skills that are often taught in separate chapters. A strong review therefore starts with classification: simplify or evaluate, solve or graph, factor or use a vertex form, apply an exponent law, preserve a rational restriction, identify a sequence pattern, or translate an application. The shortest reliable method is usually the one that matches the structure most directly.

ExpressionsLinear equationsInequalitiesSystemsSlope & linesFunctionsFactoringQuadraticsExponentsRadicalsRational expressionsSequencesPercentagesVariationAbsolute valueApplications

1. Expression work depends on signs, parentheses, and order

Negative substitutions should stay inside parentheses, while distribution must reach every term in a group.

Evaluate carefully

2 x^{2} - 3 x + 1

x = - 2

2 {(- 2)}^{2} - 3 (- 2) + 1

15

Parentheses protect the sign of the substituted value before powers are evaluated.

Simplify by structure

4 (x - 3) + 2 x - 5

4 x - 12 + 2 x - 5

6 x - 17

Distribute first and combine only matching variable parts.

2. Linear equations and inequalities share balance, but not identical sign rules

Equations preserve equality; inequalities reverse direction only when multiplication or division uses a negative value.

Linear equation

3 (x + 4) = 2 x + 11

3 x + 12 = 2 x + 11

x = - 1

Linear inequality

- 5 x + 2 \leq 17

- 5 x \leq 15

x \geq - 3

The final division is by a negative coefficient, so the inequality reverses.

3. A system solution is the point that satisfies both equations

Elimination is efficient when adding equations removes one variable immediately.

Write the system

2 x + y = 11

x - y = 1

Eliminate

3 x = 12

x = 4

Substitute back

y = 3

(4, 3)

Check the ordered pair in both original equations.

4. Slope, intercept, and a line graph describe the same linear relationship

The graph makes rise-over-run and the intercept visible before the algebra is written.

From two points to a line

(1, 3)

(5, 11)

m = \frac{11 - 3}{5 - 1}

m = 2

y = 2 x + 1

b = 1

Slope is vertical change divided by horizontal change; the constant gives the vertical intercept.

5. Function evaluation can simplify the rule without restoring excluded inputs

A canceled factor changes the formula’s appearance, but not the original domain restriction.

Simplify the function rule

f (x) = \frac{x^{2} - 9}{x - 3}

x \neq 3

f (x) = x + 3

Evaluate an allowed input

f (4) = 7

The excluded input remains excluded even though the simplified rule looks linear.

6. Factored form, vertex form, and the graph reveal different quadratic features

Factoring exposes roots, while vertex form makes the turning point visible.

One quadratic, several useful forms

x^{2} - 6 x + 5

{(x - 3)}^{2} - 4

(3, - 4)

x^{2} - 7 x + 12

(x - 1) (x - 5)

x = 1, x = 5

The graph shows the vertex and the two roots at the same time.

7. Powers and radicals simplify by exposing matching structure

Exponent laws require matching bases; radicals simplify by pulling out perfect-square factors.

Exponent laws

\frac{x^{5} \cdot x^{2}}{x^{3}}

\frac{x^{7}}{x^{3}}

x \neq 0

x^{4}

Simplify a radical

\sqrt{72}

\sqrt{36 \cdot 2}

6 \sqrt{2}

Separate the perfect-square factor from the remaining radical.

8. Rational simplification keeps every restriction from the original denominator

Factor first, write excluded values, and only then cancel common factors.

Original form

\frac{x^{2} - 16}{x^{2} + 5 x + 4}

Factor and restrict

\frac{(x - 4) (x + 4)}{(x + 1) (x + 4)}

x \neq - 4, x \neq - 1

Simplified form

\frac{x - 4}{x + 1}

The canceled denominator factor still contributes an excluded input.

9. Sequence formulas depend on the number of intervals, not just the term number

From the first term to the sixth term, the common difference is applied five times.

Arithmetic sequence

a_{n} = a_{1} + (n - 1) d

a_{1} = 7, d = 4

a_{6} = 7 + 5 \cdot 4

a_{6} = 27

The diagram makes the five intervals visible and helps prevent the common off-by-one error.

10. Percentages, variation, and short applications are modeling questions

Translate the situation first, then calculate.

Percent decrease

120 (1 - 0.15)

102

The percent is applied to the original amount.

Direct variation

y = k x

24 = k \cdot 8

k = 3

y = 3 x

Short application

18 + 6 x = 54

6 x = 36

x = 6

Define the unknown before translating the wording into an equation.

11. Absolute-value equations need both sign branches

A positive distance condition usually produces two linear equations.

Split the cases

| 2 x - 1 | = 7

2 x - 1 = 7

2 x - 1 = - 7

Keep both valid solutions

x = 4, x = - 3

Both values satisfy the required distance from the center.

12. Common high-school algebra mistakes are usually structural, not computational

A correct-looking calculation can still fail if the wrong rule was chosen or a restriction was dropped.

Unlike terms combined

Only terms with matching variable parts can be combined directly.

Sign lost during distribution

Keep negative multipliers attached to every term in the group.

Inequality direction not reversed

Reverse only when multiplying or dividing by a negative number.

Slope confused with intercept

Slope is a rate of change; the intercept is the starting vertical value.

Restriction omitted after cancellation

Original denominator zeros remain excluded from a simplified rational expression.

Wrong number of sequence intervals

The sixth term is five common differences beyond the first term.

Final high-school algebra audit

Before accepting an answer, confirm the topic, sign logic, restrictions, graph interpretation, and one quick verification step.

What problem family is this?Expression, equation, inequality, system, function, quadratic, sequence, or application?

Are signs and parentheses protected?Especially with negative substitutions, distribution, and inequalities.

Did I preserve every restriction?Rational and simplified function rules keep exclusions from the original form.

Does the algebra agree with the graph?Use slope, roots, and vertex information as an independent check.

Did I use the correct sequence interval count?Term number and number of steps are not the same quantity.

Can I verify the result quickly?Substitution, multiplication, or graph interpretation can catch many mistakes.

This block supports the High School Algebra Practice Test. Its examples are illustrative rather than copies of the test questions. The focus is simplifying and evaluating expressions, linear equations and inequalities, systems, slope, line equations, functions and restrictions, factoring, quadratics, exponents, radicals, rational expressions, arithmetic sequences, percentages, direct variation, absolute value, vertices, graphs, and short applications.