Category Archives: MAC1105

1.1 Linear Equations, Solving Equations

Higher Order Equation that reduces to a linear equation

Example: Solve the equation.

(x+5)^3-9=x(x+7)(x+8)-6

Solution:

The original equation
(x+5)^3-9=x(x+7)(x+8)-6
Simplify both sides of the equation.  On the left hand side, rewrite the exponent.  On the right hand side, begin to simplify the multiplication.
(x+5)(x+5)(x+5)-9=x(x^2+8x+7x+56)-6
Simplify both sides of the equation.  On the left hand side, begin multiplying.  On the right hand side, combine like terms.
(x+5)(x^2+5x+5x+25)-9=x(x^2+15x+56)-6
Simplify both sides of the equation.  On the left hand side, combine like terms.  On the right hand side use the distributive property.
(x+5)(x^2+10x+25)-9=x^3+15x^2+56x-6
Simplify both sides of the equation.  On the left hand side, continue multiplying.  The right hand side is in simplest form.
x(x^2+10x+25)+5(x^2+10x+25)-9=x^3+15x^2+56x-6
x^3+10x^2+25x+5x^2+50x+125-9=x^3+15x^2+56x-6
Simplify both sides of the equation.  On the left hand side, combine like terms.  The right hand side is in simplest form.
x^3+15x^2+75x+116=x^3+15x^2+56x-6
Now that each side is in simplest form we want the terms with x on one side and the constant terms on the the other side.  Subtract x^3 from each side.  It cancels from each side.
x^3-x^3+15x^2+75x+116=x^3-x^3+15x^2+56x-6
15x^2+75x+116=15x^2+56x-6
Subtract 15x^2 from each side.  It cancels from each side.
15x^2-15x^2+75x+116=15x^2-15x^2+56x-6
75x+116=56x-6
Subtract 56x from each side and simplify. 
75x-56x+116=56x-56x-6
19x+116=-6
Subtract 116 from each side and simplify.
19x+116-116=-6-116
19x=-122
Get x by it self by dividing by 19 on both sides and simplify.
{19x}/19={-122}/19
x={-122}/19
1.6 Other Types of Equations, Equations, Equations that are Quadratic in Form

Quadratic in Form (U-substitution)

Example: Solve the equation.

5x^{2/3}-6x^{1/3}+1=0

Solution:

The equation is similar to a quadratic.  It has 3 terms and one exponent is twice the other.  Since the equation is quadratic in form, use substitution to solve the equation.

Use the following substitution to rewrite the equation

u=x^{1/3}

u^2=x^{2/3}

Original Equation
5x^{2/3}-6x^{1/3}+1=0
Substitute
u=x^{1/3}
u^2=x^{2/3}
5x^{2/3}-6x^{1/3}+1=0
5u^{2}-6u+1=0
Solve the quadratic equation by factoring.
1) Factor the quadratic
5u^{2}-6u+1=0
(5u-1)(u-1)=0
Solve the quadratic equation by factoring.
2) Apply the zero product property
(5u-1)(u-1)=0
5u-1=0 or u-1=0
Solve the quadratic equation by factoring.
3) Solve each linear factor
5u-1=0 or u-1=0
5u-1+1=0+1 or u-1+1=0+1
5u=1 or u=1
{5u}/5=1/5 or u=1
u=1/5 or u=1
Substitute again to bring back the original variable.  Use the original substitution.
u=x^{1/3}
u=1/5 or u=1
x^{1/3}=1/5 or x^{1/3}=1 
Solve the equation with rational exponents.
1) Rewrite the rational exponents in radical form
x^{1/3}=1/5 or x^{1/3}=1
root{3}{x}=1/5 or root{3}{x}=1
Solve the equation with rational exponents.
2) Cancel the cube root by cubing both sides.
3) Simplify
root{3}{x}=1/5 or root{3}{x}=1
(root{3}{x})^3=(1/5)^3 or (root{3}{x})^3=(1)^3
x=1/125 or x=1

The solution to 5x^{2/3}-6x^{1/3}+1=0 isx=1/125 or x=1.

 

Here is a video with similar examples.

3.1 Functions, 3.1 Relations and Functions, Difference Quotient

Difference Quotient: Rational Function

Example:  Find the difference quotient for f(x)={4x}/{x+5}

The Difference Quotient:{f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
{f(x+h)-f(x)}/h
Write the difference quotient for the given function
={{4(x+h)}/{(x+h)+5}-{4x}/{x+5}}/h
Use the distributive property
={{4x+4h}/{x+h+5}-{4x}/{x+5}}/h
Simplify the complex fraction by multiplying the numerator and denominator by the common denominator
={{4x+4h}/{x+h+5}-{4x}/{x+5}}/h {{(x+5)(x+h+5)}/1}/ {{(x+5)(x+h+5)}/1}
Distribute the common denominator to each fraction in the numerator.
={{(4x+4h)(x+5)(x+h+5)}/{x+h+5}-{4x(x+5)(x+h+5)}/{x+5}} /{h(x+5)(x+h+5)}
Cancel the common factor
={{(4x+4h)(x+5)}-{4x(x+h+5)}} /{h(x+5)(x+h+5)}
 Multiply the expression in the numerator
={4x^2+20x+4xh+20h-4x^2-4xh-20x} /{h(x+5)(x+h+5)}
 Combine like terms
={20h} /{h(x+5)(x+h+5)}
 Cancel a common h from the numerator and denominator
={20} /{(x+5)(x+h+5)}

 

3.1 Functions, 3.1 Relations and Functions, Difference Quotient

Difference Quotient: Rational Function

Example:  Find the difference quotient for f(x)=1/x^2

The Difference Quotient:{f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
{f(x+h)-f(x)}/h
Write the difference quotient for the given function
={1/(x+h)^2-1/x^2}/h
Simplify the complex fraction by multiplying the numerator and denominator by the common denominator
={1/(x+h)^2-1/x^2}/h {{x^2(x+h)^2}/1}/{{x^2(x+h)^2}/1}
Distribute the common denominator to each fraction in the numerator.
={1/(x+h)^2-1/x^2}/h {{x^2(x+h)^2}/1}/{{x^2(x+h)^2}/1}
={{x^2(x+h)^2}/(x+h)^2-{x^2(x+h)^2}/x^2}/{h{x^2(x+h)^2}}
Cancel the common factor
={{x^2(x+h)^2}/(x+h)^2-{x^2(x+h)^2}/x^2}/{h{x^2(x+h)^2}}
={x^2-(x+h)^2}/{h{x^2(x+h)^2}}
Simplify by squaring the binomial
={x^2-(x+h)^2}/{h{x^2(x+h)^2}}
={x^2-(x+h)(x+h)}/{h{x^2(x+h)^2}}
={x^2-(x^2+xh+xh+h^2)}/{h{x^2(x+h)^2}}
Simplify by combining like terms and distributing the negative
={x^2-(x^2+2xh+h^2)}/{hx^2(x+h)^2}
={x^2-x^2-2xh-h^2}/{hx^2(x+h)^2}
 Combine like terms
={-2xh-h^2}/{hx^2(x+h)^2}
 Cancel a common h from the numerator and denominator
={-2x-h}/{x^2(x+h)^2}

The difference quotient for f(x)=1/x^2 is {-2x-h}/{x^2(x+h)^2}