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