Factoring

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Quadratic Equations

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Quadratic functions

Completing the square

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GED prep

Identifying Functions

Find domain

Find Range of a Function

Graph equations

Function Notation

Quadratic Function

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Graphing

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Quadratic Equations & Functions

Quadratic Equations & Functions

Quadratic Equation

Quadratic Equation

$x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{2a} when ax^2 + bx + c = 0$

A quadratic equation is any equation that can be rearranged in standard form as where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no term.

Standard Form:

ax^2+bx+c=0

	=	known numbers, where a ≠ 0
	=	the unknown

Quadratic formula

Quadratic Formula

With this formula you just fill in the variables from the polynomial and solve.

Example:

5x2 + 6x + 1 = 0

a = 5, b = 6, c = 1

x = −b ± √(b2 − 4ac)

x= -6 ± √(6^2 -4(5)(1)

2(5)

x = -6 ± √36 -20

x = -6 ± √16

x = −6 ± 4

the problem will be solved by taking -6 + 4 / 10 and -6 - 4 / 10

The answers would be x = -0.2 or x = -1

When the Discriminant (b²−4ac) is:

positive, there are 2 real solutions
zero, there is one real solution
negative, there are 2 complex solutions

The discriminant is the part of the quadratic formula under the square root.

Solve by factoring

Solve by factoring

Solve x² + 3x + 2 = 0

First, you have to factor x² + 3x + 2

Since the coefficient of x² is 1 (x² = 1x²), you can factor by looking for factors of the last term (last term is 2) that add up to the coefficient of the second term (3x, coefficient is 3)

2 = 1 × 2

2 = -1 × -2

1 + 2 = 3 and 3 is the coefficient of the second term.

x² + 3x + 2 = ( x + 2) × ( x + 1)

x² + 3x + 2 = 0 gives ( x + 2) × ( x + 1) = 0

( x + 2) × ( x + 1) = 0 when either x + 2 = 0 or x + 1 = 0

x + 2 = 0 when x = -2

x + 1 = 0 when x = -1

Let us now check x = -2 and x = -1 are indeed solutions of x² + 3x + 2 = 0

(-2)² + 3 × -2 + 2 = 4 + -6 + 2 = 3 + -6 = 0

(-1)² + 3 × -1 + 2 = 1 + -3 + 2 = 3 + -3 = 0

Completing the square

Solve by Completing the Square
Solve by completing the square x² + 6x + 8 = 0

x² + 6x + 8 = 0

Subtract 8 from both sides of the equation.

x² + 6x + 8 - 8 = 0 - 8

x² + 6x = - 8

You are basically looking for a term to add to x² + 6x that will make it a perfect square trinomial.

To this end, get the coefficient of the second term, divide it by 2 and raise it to the second power.

The second term is 6x and the coefficient is 6.

6/2 = 3 and after squaring 3, we get 3²

x² + 6x = - 8

Add 3² to both sides of the equation above

x² + 6x + 3² = - 8 + 3²

(x + 3)² = -8 + 9

(x + 3)² = 1

Take the square root of both sides

√((x + 3)²) = √(1)

x + 3 = ±1

When x + 3 = 1, x = -2

When x + 3 = -1, x = -4

Quadratic function

Quadratic function

A quadratic function is one of the form f(x) = ax^2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. ... A parabola intersects its axis of symmetry at a point called the vertex of the parabola.

Identify Functions

f(x)

f represents the function and; x equals the input value; therefore f(x) equals the output value

Function: A relationship where each input value has a single output value.

Domain of a function: the set of all possible input values of the function. This is the set of all the x- values

Range: the set of all possible output values for the function. This is the set of all y-values

When looking at a graph, to determine if it is a function you draw a vertical line throughout the graph and if it touches more than one point on the graph than it is not considered a function.