Algebra Test 1
a²+b²
(a+bi)(a-bi)=?
The equation to solve a perfect square
(b/2)²
Steps required to convert from Standard form of a circle to General form.
(x-3)²+(y+4)²=17 Step 1: Factor x²-6x+9+y²+8y+16=17 Step 2: Rearrange into general form x²+y²-6x+8y+8=0
Equation of a Circle
(x-h)²+(y-k)²=r²
Steps to a word problem
1. Read the problem 2. Have a plan: table, diagram, or some form of organization 3. Establish the equation and solve
Conditional Equation
An equation with a finite number of solutions
Inconsistent Equation
An equation with no solution
General Form of a Line
Ax+By=C
Equation for Celsius
C=5/9(F-32)
√(x₂-x₁)²+(y₂-y₁)²
Distance Formula: This the distance between two points
(x-h)²+(y-k)²=r²
Equation of a circle
Work Rate X Time = Work Completed
Equation to a work problem
Identity
Every real # is a solution
Multiply by the conjugate
How do you solve the problem: (3+4i)/(7-5i)
m₁
If line l₁‖l₂ then m₂=?
-(1/m₁)
If line l₁⊥l₂ then m₂=?
Reverse the signs
In order to find (h,k) from the standard form of a circle, you must do what to h and k?
Add the perfect square to both sides
In order to find the center and radius of a circle that is not in standard form, such as x^2+bx, you must do what?
Take the square root
In order to find the r of a circle when the equation is standard form, what must you do to r?
Step 1: Isolate the most complicated radical √(2x+1)=1+√x Step 2: Square both sides (√2x+1)²=(√x+1)² 2x+1=x+2√(x)+1 Step 3: Simplify and regroup x=2√x Step 4: Square both sides x²=4x ---> x²-4x=0 Step 5: Factor the remaining quadratic x(x+4)=0 Step 6: Use the zero product property to solve x=0 and x+4=0 x=-4
List the steps in isolating a radical. Use the following equation: √(2x+1)-√x=1
(x₁+x₂/2 , y₁+y₂/2)
Midpoint Formula: This finds the exact midpoint between two points
y-y₁=m(x-x₁)
Point Slope form: if you know a point and the slope, then use this form.
Three equations of a line
Point-Slope, Slope-Intercept, and General Form
a²+b²=c²
Pythagorean theorem: This equation determines the hypotenuse of a triangle
x= (-b±√(b²-4ac))/2a
Quadratic Formula
m=(y₂-y₁)/(x₂-x₁)
Slope Formula
y=mx+b
Slope Intercept Form
Step 1: Grouping (x³+x²)+(5x+5)=0 Step 2: Factor x²(x+1)+5(x+1)=0 Step 3: Regroup by factors (x²+5)(x+1)=0 Step 4: Solve using the Zero Product Property x²+5=0 and x+1=0 x²=-5 and x=-1 x=±√-5 x=±i√5
Solve the cubic equation: x³+x²+5x+5=0
Step 1: Put into standard form (ax²+bx+c=0) 3x²-6x+7=0 Step 2: Put into the quadratic equation x=(-6±√(-6)²-4(3)(7))/(2(3) Step 3: Simplify x=(-6±√120)/6 Step 4 Simplify Discriminant x=(-6±2√30)/6 Step 5 Further simplify until finished x=3±√30
Solve the equation using the quadratic formula: 3x²-6x=-7
(2x-1)(x+1)
Solve the following equation by factoring 2x²+x-1=0
-2(x+3)²=-10 (x+3)²=5 x+3=±√5 x=-3±√5
Solve the following equation using the square root property: -2(x+3)²+10=0
x=±√-4 x=±i√4 x=±2i
Solve the following equation using the square root property: x²=-4
Step 1: Factor out a variable to get to a sum of cubes (a³+b³) 2x(x³+8x)=0 Step 2: Put into the Sum of Cubes equation: 2x(x+2)(x²-2x+2²)=0 Step 3: Use the zero product property: 2x=0 and x+2=0 and x²-2x+4=o x=0 and x=-2 and use not solved Step 4: Solve the remaining equation using the quadratic formula and simplify. x=(2±√-12)/2 Step 5: Solve x=1±i√3
Solve the following equation using the sum of cubes equation: 2x⁴+16x=0
Step 1: Note that the second exponent is half of the first exponent. Therefore this is a "quadratic type equation" and can be solved factoring. (x²-7)(x²+4)=0 Step 2: Use the zero product property to solve x²-7=0 and x²+4=0 x²=7 and x²=-4 Step 3: Use the square root property x=±√7 and x=±√-4 x=±2i
Solve the following equation: x⁴-3x²-28=0
Step 1: Change to a positive exponent by taking the inverse 1/(x³/²)=1/27 Step 2: Cross multiply x³/²=27 Step 3: Solve by setting both sides equal to the index of the radical and then simplify x³=27² x³=729 Step 4:Take the cube root of both sides (remember that a ± is not needed as the index is even.) x=³√729 x=9
Solve x-³/²=1/27
Step 1: Set both sides equal to the index of the radical x²=4³ x²=64 Step 2: Use the square root property to solve x=±√64 Step 3: Simplify if needed x=±8
Solve x²/³=4
2i
Solve √(-4)
x²+y²+Dx+Ey+F=0
The General Form of the Equation of a Circle
(b/2)²
The equation for a perfect square where b=the second integer. Add this to both sides to complete the square.
Amount X Concentration = Amount of substance
The equation to a mixture question.
R x T = D
The equation to a uniform motion problem
No real solutions
The equation x²+1=0 has how many real solutions?
Use the distance formula to determine radius
The first step or finding the equation of a circle given its center and a point on its circumference.
Radius
The fixed distance from a circle's center to any point on the circle
(h,k)
The point which is the center of a circle
Zero Product Property
The property such that if the product of two real number's is 0, then one must be 0.
Circle
The set of all points in a plane that are equidistant from a fixed point in the center
Equal
The slope of parallel lines are what?
Factoring, Square Root Property, Completing the Square, Quadratic Formula
These are the four methods for solving a quadratic equation.
ax²+bx+c=0
This is the standard form for a quadratic equation
Slope Formula
This measures the steepness of a line with respect to the x axis
Mixture Question
This type of question can also be seen as a percentage, or rate such as tax/commission
x=±√k
Using the square root property, you can determine that x²=k is also equal to what?
Step 1: Leading coefficient of 1 Step 2: Isolate x variables x²-4x=-1/3 Step 3: Complete the square x²-4x+4=-1/3 + 4 ---> (x-2)²=-1/3+4 Step 4: Use the square root property x-2=±√-11/3 ---> x=2±i√11/3) Step 5: Rationalize if needed x=2±i√(33)/3
What are the steps in solving a quadratic by completing the square? Use the equation of 3x²-12x+1=0 as reference
Sum of cubes
What does the following equation represent? (a³+b³)=(a+b)(a²-ab+b²)
a+bi
What is the standard form of complex number?
(a³+b³)=(a+b)(a²-ab+b²)
What is the sum of cubes equation?
Check for extraneous solutions
When factoring radicals, what is the very last step?
(a+bi)(a-bi)
a²+b²=?
Pythagorean Theorem
a²+b²=c²
y₁-mx₁
b in Slope Intercept form y=mx+b
Pythagorean Theorem C=
c=√(a²+b²)
√-1
i=?
-1
i²=?
i⁰=0, i¹=i, i²=-1, i³-i, i⁴=1
i⁰, i¹, i², i³, i⁴ each equal what?
Steps required to convert from General Form to Standard form of a circle.
x²+y²+4x-4y-1=0 Step 1: Regroup (x²+4x) +(y²-4y) = 1 Step 2: Complete the square (x²+4x+4)+(y²-4y+4)=1+4+4 Step 3: Factor (x+2)²+(y-2)²=9 --> (x+2)²+(y-2)²=3² Step 4: Find Center and R Center=(2,-2) and r=3
General Form of the Equation of a Circle
x²+y²+Dx+Ey+F=0
Square Root Property
x²=k is equal to x=±√k
Point Slope Form
y-y₁=m(x-x₁)
Slope Intercept Form
y=mx+b
b in Slope intercept form
y₁-mx₁