Education
Maple 2024 includes a number of improvements to support teaching and learning ofmathematics and science.

Step-by-Step Solutions
Maple 2024 improves the existing suite of commands for showing step-by-step solutions tostandard math problems. It also adds some new methods as follows:

Partial Fraction Decomposition
Partial Fraction Decomposition is a method used in algebra and calculus to simplify
complex fractions or rational expressions into a sum of simpler fractions.
Maple can show you the steps required to perform partial fraction decomposition withthe new command PartialFractionSteps.

with Student:-Basics :PartialFractionSteps LL•2xxC3 xC2

Use the factors to write an equation for the partial fraction decompositionL•

4 x2 C 4 x
2 x2 C 8 x C 6 x C 2First, factor the numerator and denominator. If we can simplify by dividing terms, we do so.
L•

2xxC3 xC2=

AB
CxC2xC3

Multiply by the LHS's denominator.
L2 x =•

4 x2 C 4 x
2 x2 C 8 x C 6 $ x C 2ABCxC2xC3xC3xC2

Expand everything out.
L2 x = A x C B x C 3 A C 2 B•

Equate the coefficients to create equations
0 = 3 A C 2 B, L2 = A C B•

Solve the equations.
A = 4, B = L6

•

Use these values in the partial fraction decomposition and simplify.
The final partial fraction decomposition isL

64
CxC3xC2Solve Steps
Maple 2024 improves the steps given by the SolveSteps command.
The following example recognizes ln x2 as a common factor and goes into greater detailwhen applying log rules.

with Student:-Basics :SolveSteps ln x2 = 2$ln x222;

Let's solve
ln x222ln x2
22ln x2
L22ln x2 =•Simplify
ln x2 L•Factor2ln x•=0Remove rationals and common factor
L

•

=0

1$ln x2 $ L2 C ln x2
222
The 1st factor is ln x which implies ln x = 0 is a solution
ln x2 = 0•

2Set 2nd factor L2 C ln x to 0 to solve
L2 C ln x2 = 0•2Solution of L2 C ln x = 0ln x2 = 2•Solutionln x2 = 0, 2•Consider 1st solution
ln x2 = 0•Raise both sides to a power of e2eln x = e0•

ln X
Log rule: e=Xx2 = e0

•Simplify
x2 = 1•

Take Square root of both sides
x= G•

1Consider 2nd solution
ln x2 = 2•

Raise both sides to a power of e2eln x = e2
•ln X
Log rule: e=Xx2 = e2•

Take Square root of both sides
x= G

e2

Check if x = L1 satisfies domain requirements
2
Domain requirement from ln x , cannot take log of a negative number0 % x2
2
Substitute x = L1 into x and evaluate
L1 2
Evaluate exponents
1
Domain requirement met
0%1
Therefore the domain requirement is satisfied.
#•

Substitute solution into equation and check if left-side = right-side
0=0•

left-side = right-side
0=0•

x = L1 satisfies all the domain requirements and left-side = right-side, it is a solution#

Check if x = 1 satisfies domain requirements
2
Domain requirement from ln x , cannot take log of a negative number
0 % x2
2
Substitute x = 1 into x and evaluate
12
Domain requirement met
0%1
Therefore the domain requirement is satisfied.
#•

Substitute solution into equation and check if left-side = right-side
0=0•

left-side = right-side
0=0•x = 1 satisfies all the domain requirements and left-side = right-side, it is a solution#
Check if x =2
e satisfies domain requirements

2
Domain requirement from ln x , cannot take log of a negative number
0 % x2
Substitute x =e222
e into x and evaluate

2

Evaluate exponents
e2
Evaluate
7.389056099
Domain requirement met
0 % 7.389056099
Therefore the domain requirement is satisfied.
#•

Substitute solution into equation and check if left-side = right-side
2=2•

left-side = right-side
2=2

•

x=

2
e satisfies all the domain requirements and left-side = right-side, it is a solution

2
e satisfies all the domain requirements and left-side = right-side, it is a solution

x=#

Check if x = L

2
e satisfies domain requirements

2
Domain requirement from ln x , cannot take log of a negative number
0 % x222
Substitute x = L e into x and evaluate
2L e2

Evaluate exponents
e2Evaluate
7.389056099
Domain requirement met
0 % 7.389056099
Therefore the domain requirement is satisfied.
#•

Substitute solution into equation and check if left-side = right-side
2=2•

left-side = right-side
2=2

•

x=L

2
e satisfies all the domain requirements and left-side = right-side, it is a solution#•

Exact solutions
x = L1, 1,•

e2 , L e2

Approximate solutions
x = L2.718281828, L1., 1., 2.718281828

In the example below, the step-by-step solution uses rules to handle logs of constants
that are perfect powers. Note that we use the inert-form name %log instead of log inorder to prevent evaluation.

with Student:-Basics :SolveSteps%log 11 25%log 11 5
Let's solve=6x ;Let's solve
log11 25
log11 5•

= 6$x

Rearrange expression
6$x =log11 25
log11 52
2
Rewrite log 25 as log
5 , then apply the log rule log
5 = 2 log 511111111•

6x = 2•Divide both sides by 66$x
2
=66

•

Simplify
x=13

When solving systems of equations such as the following, it is important to select theright variable to start with. SolveSteps analyzes the system and chooses an appropriate
variable to start with, thus avoiding more complicated steps.

with Student:-Basics :SolveStepsy =Lx2 C 6 x L 10, y = 2 x2 C 6 x C 5 ;Let's solve
y = L1 $x2 C 6$x L 10, y = 2$x2 C 6$x C 5•

E1: Pick the 2nd equation which is already solved for yy = 2$x2 C 6$x C 5•

Pick the 1st equation to solve for x
y = L1 $x2 C 6$x L 10•

2
Substitute the value of y = 2 x C 6 x C 5 into the 1st equation of the system
2 x2 C 6 x C 5 = L1 $x2 C 6$x L 10
Isolate for x
2Add x L 6 x C 10 to both sides
2 x2 C 6 x C 5 C x2 L 6 x C 10 = Lx2 C 6 x L 10 C x2 L 6 x C 10
Simplify
3 x2 C 15 = 0
Subtract 15 from both sides
3$x2 C 15 L 15 = 0 L 15
Simplify

3$x2 = L15
Divide both sides by 3
L15
3$x2
=
33
Simplifyx2 = L5Take Square root of both sides
x= GL5No real solutions
x = imaginary
Maple 2024 can now give steps in more situations involving expressions with absolutevalues. For example,with Student:-Basics :SolveSteps 2 x C 3 R 2 ;Check My Work: Help with Math Problems
Building on the suite of SolvePractice / SolveFeedback commands, Maple 2024 introduces
new commands to help identify where mistakes occur in worked problems relating tofactoring, simplification, and limits.

Grading:-FactorPractice 2$x3 L 8$x

Check My Work> New Math Entry Box

Grading:-SimplifyPractice cos x 3 C cos x sin x 2

Check My Work> New Math Entry Box

Grading:-LimitPractice Limit

sin x,x=0x;Check My Work> New Math Entry Box

New Math Apps
Two Math Apps have been added to our collection.

New MathApp

Description

The Tangent Planes app explores surfaces defined byfunctions of two variables and shows the connection
Tangent Planes

between a function's partial derivatives and thesurface's tangent plane at a point.

The Golden Angle app offers a visualization of thepatterns formed by rotations of a given angle around
Golden Anglea circle. The resulting pattern, using the golden angleitself, mimics patterns found in nature.