Subjects

Algebra 2

Rational & Root Equations

Quadratic Systems

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Jun 12, 2023

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1 page

Easy Guide: Convert Standard Form to Vertex Form & Learn Solving Quadratics!

Name: Knowunity
Brand: Knowunity

Mira

@miraa._noui

Converting between standard form and vertex form of parabolas is... Show more

final study guide
parabolas
standard form: ax² + bx + C
→vertex form: a (x-n)² + k
→ intercept form: a (x-p)(x-9).
•vertex:
1.-/29 for x-coo

Parabolas and Quadratic Equations Study Guide

This comprehensive guide covers essential concepts related to parabolas and quadratic equations, focusing on different forms of quadratic expressions, conversion methods, and solving techniques.

Definition: A parabola is a U-shaped curve that can be represented by a quadratic equation.

Forms of Quadratic Expressions

Quadratic expressions can be written in three main forms:

Standard form: ax² + bx + c
Vertex form: a $x-h$ ² + k
Intercept form: a $x-p$ $x-q$

Highlight: The vertex form is particularly useful for identifying the parabola's turning point and axis of symmetry.

Vertex of a Parabola

The vertex of a parabola can be found using two methods:

x-coordinate: -b/ $2a$ $also the axis of symmetry$
$h,k$ in vertex form

Example: If a > 0, the vertex is a minimum point; if a < 0, the vertex is a maximum point.

Finding x-intercepts

X-intercepts can be determined by:

Factoring the quadratic expression
Using the quadratic formula or completing the square

Completing the Square

Completing the square is a method used to convert from standard form to vertex form and solve quadratic equations. The process involves the following steps:

Add $b/2a$ ² to both sides of the equation
Factor the left side and simplify the right side
Solve for intercepts by taking the square root of both sides and subtracting the extra term

Vocabulary: Completing the square is a technique used to rewrite a quadratic expression as a perfect square trinomial plus a constant.

Converting to Vertex Form

To convert a quadratic expression from standard form to vertex form:

Rewrite as: a $x² + (b/a$ x) + c
Complete the square: a $x² + (b/a$ x + $b/2a$ ²) + c - a $b/2a$ ²
Factor: a $x + b/2a$ ² + $c - b²/4a$

Highlight: The vertex form a $x-h$ ² + k is derived from completing the square.

The Quadratic Formula

The quadratic formula is used to solve quadratic equations:

x = $-b ± √(b²-4ac$ ) / $2a$

Definition: The discriminant is the expression under the square root in the quadratic formula: b²-4ac.

Understanding the Discriminant

The discriminant helps determine the nature of a quadratic equation's roots:

Positive: Two real solutions
Perfect square: Two rational solutions
Zero: One real solution $the vertex$
Negative: No real solutions $two imaginary solutions$

Example: For the equation x² + 4x + 4 = 0, the discriminant is 4² - 4 $1$ $4$ = 0, indicating one real solution.

Systems of Quadratic Equations

When solving systems involving quadratic equations:

Graph to check for intersections
Only consider real solutions

Quadratic Inequalities

When solving quadratic inequalities:

Choose a test point not on the parabola to determine which region to shade
For inequalities with two variables, perform a point test for both equations and shade the shared regions

Highlight: The point test is crucial for determining the solution region of quadratic inequalities.

This study guide provides a comprehensive overview of parabolas and quadratic equations, covering essential concepts and techniques for solving various problems related to these topics.

Students love us — and so will you.

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Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

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In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

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Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

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Subjects

Algebra 2

Rational & Root Equations

Quadratic Systems

Algebra 2

•

469

•

Jun 12, 2023

•

1 page

Easy Guide: Convert Standard Form to Vertex Form & Learn Solving Quadratics!

Mira

@miraa._noui

Converting between standard form and vertex form of parabolas is a crucial skill in algebra. This guide covers key concepts related to parabolas, including standard to vertex form conversion, completing the square, and understanding the discriminant in quadratic... Show more

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Parabolas and Quadratic Equations Study Guide

This comprehensive guide covers essential concepts related to parabolas and quadratic equations, focusing on different forms of quadratic expressions, conversion methods, and solving techniques.

Definition: A parabola is a U-shaped curve that can be represented by a quadratic equation.

Forms of Quadratic Expressions

Quadratic expressions can be written in three main forms:

Standard form: ax² + bx + c

Vertex form: a $x-h$ ² + k

Intercept form: a $x-p$ $x-q$

Highlight: The vertex form is particularly useful for identifying the parabola's turning point and axis of symmetry.

Vertex of a Parabola

The vertex of a parabola can be found using two methods:

x-coordinate: -b/ $2a$ $also the axis of symmetry$

$h,k$ in vertex form

Example: If a > 0, the vertex is a minimum point; if a < 0, the vertex is a maximum point.

Finding x-intercepts

X-intercepts can be determined by:

Factoring the quadratic expression

Using the quadratic formula or completing the square

Completing the Square

Completing the square is a method used to convert from standard form to vertex form and solve quadratic equations. The process involves the following steps:

Add $b/2a$ ² to both sides of the equation

Factor the left side and simplify the right side

Solve for intercepts by taking the square root of both sides and subtracting the extra term

Vocabulary: Completing the square is a technique used to rewrite a quadratic expression as a perfect square trinomial plus a constant.

Converting to Vertex Form

To convert a quadratic expression from standard form to vertex form:

Rewrite as: a $x² + (b/a$ x) + c

Complete the square: a $x² + (b/a$ x + $b/2a$ ²) + c - a $b/2a$ ²

Factor: a $x + b/2a$ ² + $c - b²/4a$

Highlight: The vertex form a $x-h$ ² + k is derived from completing the square.

The Quadratic Formula

The quadratic formula is used to solve quadratic equations:

x = $-b ± √(b²-4ac$ ) / $2a$

Definition: The discriminant is the expression under the square root in the quadratic formula: b²-4ac.

Understanding the Discriminant

The discriminant helps determine the nature of a quadratic equation's roots:

Positive: Two real solutions

Perfect square: Two rational solutions

Zero: One real solution $the vertex$

Negative: No real solutions $two imaginary solutions$

Example: For the equation x² + 4x + 4 = 0, the discriminant is 4² - 4 $1$ $4$ = 0, indicating one real solution.

Systems of Quadratic Equations

When solving systems involving quadratic equations:

Graph to check for intersections

Only consider real solutions

Quadratic Inequalities

When solving quadratic inequalities:

Choose a test point not on the parabola to determine which region to shade

For inequalities with two variables, perform a point test for both equations and shade the shared regions

Highlight: The point test is crucial for determining the solution region of quadratic inequalities.

This study guide provides a comprehensive overview of parabolas and quadratic equations, covering essential concepts and techniques for solving various problems related to these topics.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Easy Guide: Convert Standard Form to Vertex Form & Learn Solving Quadratics!

Parabolas and Quadratic Equations Study Guide

Forms of Quadratic Expressions

Vertex of a Parabola

Finding x-intercepts

Completing the Square

Converting to Vertex Form

The Quadratic Formula

Understanding the Discriminant

Systems of Quadratic Equations

Quadratic Inequalities

Similar Content

8.2 Graph and write equations of parabolas

8.3 Equations of circle

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

4.8/5

Easy Guide: Convert Standard Form to Vertex Form & Learn Solving Quadratics!

Sign up to see the contentIt's free!

Parabolas and Quadratic Equations Study Guide

Forms of Quadratic Expressions

Vertex of a Parabola

Finding x-intercepts

Completing the Square

Converting to Vertex Form

The Quadratic Formula

Understanding the Discriminant

Systems of Quadratic Equations

Quadratic Inequalities

Similar Content

8.2 Graph and write equations of parabolas

8.3 Equations of circle

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

4.8/5