You can solve a quadratic by factoring, square roots or the quadratic formula, and sketch the parabola it makes.
Weeks 13-14
Algebra I course units
Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.
01
Linear equations
Weeks 1-2
You can solve an equation for one unknown even when it has parentheses or letters on both sides. You can also put your answer back in to prove it works.
Solve 3(x - 4) = 2x + 5 by clearing the parentheses first.
Clear fractions out of x/3 + 2 = 7 by multiplying every term by 3.
Rearrange a formula such as P = 2L + 2W to get W by itself.
You can solve an inequality and show the answer on a number line or in interval form. You know the one move that flips the inequality sign around.
Solve -2x + 1 > 9 and flip the sign when you divide by a negative.
Graph x >= -3 with a filled dot and an arrow to the right.
Turn 'at most 40 hours' into h <= 40 and solve a pay problem with it.
Worked example
Solve -2x + 1 > 9
-2x > 8
Divide by -2 and flip the sign
Answerx < -4 (check x = -5: -2(-5) + 1 = 11, and 11 > 9)
Most common mistakeKeeping the sign the same after dividing by a negative: answering x > -4, which wrongly accepts x = 0 even though -2(0) + 1 = 1 is not greater than 9.
You can decide whether a rule is a function, read values off its graph, and use notation like f(3). You can also state the domain and range in plain words.
Run the vertical line test on a graph to decide if it is a function.
Evaluate f(x) = x squared - 2x at x = 5.
State that the domain of f(x) = 1/(x - 3) is every number except 3.
Worked example
If f(x) = x squared - 2x, find f(5)
Replace every x with 5: f(5) = 5 squared - 2(5)
= 25 - 10
Answerf(5) = 15
Most common mistakeReading f(5) as f times 5: multiplying the whole rule by 5 to get 5x squared - 10x instead of replacing the x's and getting the single number 15.
You can find the point where two lines cross by graphing, substitution or elimination. You can also spot when two lines never cross or are secretly the same line.
Solve by substitution when one equation already has y by itself.
Add two equations to cancel a variable, as with 2x + y = 7 and 3x - y = 8.
Read a final line of 0 = 5 as a sign that there is no solution at all.
Worked example
Solve 2x + y = 7 and 3x - y = 8
Add the two equations so y cancels: 5x = 15
x = 3
AnswerPut x = 3 back in: 6 + y = 7, so the answer is (3, 1)
Most common mistakeAdding the left sides but forgetting the right sides: writing 5x = 7 and reporting x = 1.4 instead of x = 3.
You can multiply, divide and stack powers without ever writing them out long, including zero and negative exponents. You can also write very large and very small numbers in scientific notation.
Simplify x cubed times x to the fifth as x to the eighth by adding exponents.
Rewrite 2 to the power -3 as 1/8.
Write 0.00042 as 4.2 x 10 to the power -4.
Worked example
Simplify (2x cubed) to the fourth power
Raise both parts: 2 to the fourth times x cubed to the fourth
16 times x to the twelfth
Answer16x to the twelfth power
Most common mistakeRaising only the variable: writing 2x to the twelfth instead of 16x to the twelfth, because the 2 out front was never raised to the fourth power.
You can solve a quadratic by factoring, square roots or the quadratic formula, and sketch the parabola it makes. You can also find the vertex and say whether it is the high point or the low point.
Solve x squared - 2x - 15 = 0 by factoring into (x - 5)(x + 3).
Use the quadratic formula on 2x squared + 3x - 2 = 0.
Find the vertex of y = x squared - 6x + 5 with x = -b/(2a).
Worked example
Find the vertex of y = x squared - 6x + 5
x = -b/(2a) = 6/2 = 3
y = 9 - 18 + 5 = -4
AnswerThe vertex is (3, -4), and it is the lowest point
Most common mistakeDropping the minus in -b/(2a): using x = -6/2 = -3, which lands on the point (-3, 32), far up the other arm of the parabola.
Before timing yourself, check whether you can explain Linear equations from a blank page. Then connect it to Inequalities. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Algebra calculator can test calculations and representations used in Algebra I. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Algebra I practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Linear equations if you are following the full sequence. If that unit feels automatic, open the Algebra I practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Algebra I test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Algebra calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
