February 23, 2019  February 23, 2019 07:00 AM PST 09:00 AM PST Learn reading strategies that can help even non-voracious reader to master GMAT RC. Saturday, February 23rd at 7 AM PT February 24, 2019  February 24, 2019 07:00 AM PST 09:00 AM PST Get personalized insights on how to achieve your Target Quant Score.
Author |
Message |
TAGS:
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 53067
|
Algebra: Tips and hints
[#permalink]
Show Tags
16 Jul 2014, 10:30
Algebraic Identities1. \((x+y)^2=x^2+y^2+2xy\) 2. \((x-y)^2=x^2+y^2-2xy\) 3. \(x^2-y^2=(x+y)(x-y)\) 4. \((x+y)^2-(x-y)^2=4xy\) 5. \(x^3+y^3=(x+y)(x^2+y^2-xy)\) 6. \(x^3-y^3=(x-y)(x^2+y^2+xy)\) QuadraticsThe general form of a quadratic equation is \(ax^2+bx+c=0\). It's roots are: \(x_1=\frac{-b-\sqrt{b^2-4ac}}{2a}\) and \(x_2=\frac{-b+\sqrt{b^2-4ac}}{2a}\) Expression \(b^2-4ac\) is called discriminant: - If discriminant is positive quadratics has two roots;
- If discriminant is negative quadratics has no root;
- If discriminant is zero quadratics has one root.
When graphed quadratic expression (\(ax^2+bx+c=0\)) gives parabola:  - The larger the absolute value of \(a\), the steeper (or thinner) the parabola is, since the value of y is increased more quickly.
- If \(a\) is positive, the parabola opens upward, if negative, the parabola opens downward.
Viete's theoremViete's theorem states that for the roots \(x_1\) and \(x_2\) of a quadratic equation \(ax^2+bx+c=0\):
\(x_1+x_2=\frac{-b}{a}\) AND \(x_1*x_2=\frac{c}{a}\).Common mistake to avoidNever reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.For example, \(xy=y\) cannot be reduced by \(y\) because \(y\) could be 0 and we cannot divide by 0. If we do we'll loose one of the solutions. The correct way is: \(xy=y\) --> \(xy-y=0\) --> \(y(x-1)=0\) --> \(y=0\) or \(x=1\). Please share your Algebra tips below and get kudos point. Thank you.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
 What are GMAT Club Tests? Extra-hard Quant Tests with Brilliant Analytics
|
|
|
Manager
Joined: 20 Dec 2011
Posts: 73
|
Re: Algebra: Tips and hints
[#permalink]
Show Tags
17 Jul 2014, 13:28
Bunuel wrote: Algebraic Identities 3. \(x^2-y^2=(x+y)(x-y)\) Rule 3 is especially useful on GMAT. Sometimes it is obvious, as in PS 117 in OG 13: if-n-3-8-2-8-which-of-the-following-is-not-a-factor-of-n-132874.htmlbut sometimes it will be hidden, as in PS 199 in OG 13: topic-137149.htmlIn other words, if you are stuck and you see anything that might be expressed as "[perfect square] - [perfect square]", see if this can help you.
|
|
|
Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 273
|
Algebra: Tips and hints
[#permalink]
Show Tags
Updated on: 20 Sep 2017, 09:34
Bunuel wrote: Algebraic Identities1. \((x+y)^2=x^2+y^2+2xy\) 2. \((x-y)^2=x^2+y^2-2xy\) 3. \(x^2-y^2=(x+y)(x-y)\) 4. \((x+y)^2-(x-y)^2=4xy\) 5. \(x^3+y^3=(x+y)(x^2+y^2-xy)\) 6. \(x^3-y^3=(x-y)(x^2+y^2+xy)\) QuadraticsThe general form of a quadratic equation is \(ax^2+bx+c=0\). It's roots are: \(x_1=\frac{-b-\sqrt{b^2-4ac}}{2a}\) and \(x_2=\frac{-b+\sqrt{b^2-4ac}}{2a}\) Expression \(b^2-4ac\) is called discriminant: - If discriminant is positive quadratics has two roots;
- If discriminant is negative quadratics has no root;
- If discriminant is zero quadratics has one root.
When graphed quadratic expression (\(ax^2+bx+c=0\)) gives parabola:  - The larger the absolute value of \(a\), the steeper (or thinner) the parabola is, since the value of y is increased more quickly.
- If \(a\) is positive, the parabola opens upward, if negative, the parabola opens downward.
Viete's theoremViete's theorem states that for the roots \(x_1\) and \(x_2\) of a quadratic equation \(ax^2+bx+c=0\):
\(x_1+x_2=\frac{-b}{a}\) AND \(x_1*x_2=\frac{c}{a}\).Common mistake to avoidNever reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.For example, \(xy=y\) cannot be reduced by \(y\) because \(y\) could be 0 and we cannot divide by 0. If we do we'll loose one of the solutions. The correct way is: \(xy=y\) --> \(xy-y=0\) --> \(y(x-1)=0\) --> \(y=0\) or \(x=1\). Please share your Algebra tips below and get kudos point. Thank you. hi man great you are .. I want to know parabola... please let me understand few things as under: 1. When graphed quadratic expression (ax^2 + bx + c= 0) gives parabola: Perhaps, on the graph, you plotted a, b, and c, please say to me which one is which ...? 2. The larger the absolute value of a, the steeper (or thinner) the parabola is, since the value of y is increased more quickly: please shed some light on this concept .... 3. If a is positive, the parabola opens upward, if negative, the parabola opens downward. also, shed some light ... maybe they are very obvious....but I need some sort of clarification and your help... thanks in advance, man..
Originally posted by testcracker on 20 Sep 2017, 09:10.
Last edited by testcracker on 20 Sep 2017, 09:34, edited 1 time in total.
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 53067
|
Re: Algebra: Tips and hints
[#permalink]
Show Tags
20 Sep 2017, 09:15
gmatcracker2017 wrote: Bunuel wrote: Algebraic Identities1. \((x+y)^2=x^2+y^2+2xy\) 2. \((x-y)^2=x^2+y^2-2xy\) 3. \(x^2-y^2=(x+y)(x-y)\) 4. \((x+y)^2-(x-y)^2=4xy\) 5. \(x^3+y^3=(x+y)(x^2+y^2-xy)\) 6. \(x^3-y^3=(x-y)(x^2+y^2+xy)\) QuadraticsThe general form of a quadratic equation is \(ax^2+bx+c=0\). It's roots are: \(x_1=\frac{-b-\sqrt{b^2-4ac}}{2a}\) and \(x_2=\frac{-b+\sqrt{b^2-4ac}}{2a}\) Expression \(b^2-4ac\) is called discriminant: - If discriminant is positive quadratics has two roots;
- If discriminant is negative quadratics has no root;
- If discriminant is zero quadratics has one root.
When graphed quadratic expression (\(ax^2+bx+c=0\)) gives parabola:  - The larger the absolute value of \(a\), the steeper (or thinner) the parabola is, since the value of y is increased more quickly.
- If \(a\) is positive, the parabola opens upward, if negative, the parabola opens downward.
Viete's theoremViete's theorem states that for the roots \(x_1\) and \(x_2\) of a quadratic equation \(ax^2+bx+c=0\):
\(x_1+x_2=\frac{-b}{a}\) AND \(x_1*x_2=\frac{c}{a}\).Common mistake to avoidNever reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.For example, \(xy=y\) cannot be reduced by \(y\) because \(y\) could be 0 and we cannot divide by 0. If we do we'll loose one of the solutions. The correct way is: \(xy=y\) --> \(xy-y=0\) --> \(y(x-1)=0\) --> \(y=0\) or \(x=1\). Please share your Algebra tips below and get kudos point. Thank you. hi man great you are .. I want to know parabola... please let me understand few things as under: 1. When graphed quadratic expression (ax^2 + bx + c= 0) gives parabola: Perhaps, on the graph, you presented a, b, and c, please say to me which one is which ...? 2. The larger the absolute value of a, the steeper (or thinner) the parabola is, since the value of y is increased more quickly: please shed some light on this concept .... 3. If a is positive, the parabola opens upward, if negative, the parabola opens downward. also, shed some light ... maybe they are very obvious....but I need some sort of clarification... thanks in advance, man.. Check here: http://www.mathopenref.com/quadraticexplorer.html
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
 What are GMAT Club Tests? Extra-hard Quant Tests with Brilliant Analytics
|
|
|
Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 273
|
Re: Algebra: Tips and hints
[#permalink]
Show Tags
20 Sep 2017, 13:29
Bunuel wrote: gmatcracker2017 wrote: Bunuel wrote: Algebraic Identities1. \((x+y)^2=x^2+y^2+2xy\) 2. \((x-y)^2=x^2+y^2-2xy\) 3. \(x^2-y^2=(x+y)(x-y)\) 4. \((x+y)^2-(x-y)^2=4xy\) 5. \(x^3+y^3=(x+y)(x^2+y^2-xy)\) 6. \(x^3-y^3=(x-y)(x^2+y^2+xy)\) QuadraticsThe general form of a quadratic equation is \(ax^2+bx+c=0\). It's roots are: \(x_1=\frac{-b-\sqrt{b^2-4ac}}{2a}\) and \(x_2=\frac{-b+\sqrt{b^2-4ac}}{2a}\) Expression \(b^2-4ac\) is called discriminant: - If discriminant is positive quadratics has two roots;
- If discriminant is negative quadratics has no root;
- If discriminant is zero quadratics has one root.
When graphed quadratic expression (\(ax^2+bx+c=0\)) gives parabola:  - The larger the absolute value of \(a\), the steeper (or thinner) the parabola is, since the value of y is increased more quickly.
- If \(a\) is positive, the parabola opens upward, if negative, the parabola opens downward.
Viete's theoremViete's theorem states that for the roots \(x_1\) and \(x_2\) of a quadratic equation \(ax^2+bx+c=0\):
\(x_1+x_2=\frac{-b}{a}\) AND \(x_1*x_2=\frac{c}{a}\).Common mistake to avoidNever reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.For example, \(xy=y\) cannot be reduced by \(y\) because \(y\) could be 0 and we cannot divide by 0. If we do we'll loose one of the solutions. The correct way is: \(xy=y\) --> \(xy-y=0\) --> \(y(x-1)=0\) --> \(y=0\) or \(x=1\). Please share your Algebra tips below and get kudos point. Thank you. hi man great you are .. I want to know parabola... please let me understand few things as under: 1. When graphed quadratic expression (ax^2 + bx + c= 0) gives parabola: Perhaps, on the graph, you presented a, b, and c, please say to me which one is which ...? 2. The larger the absolute value of a, the steeper (or thinner) the parabola is, since the value of y is increased more quickly: please shed some light on this concept .... 3. If a is positive, the parabola opens upward, if negative, the parabola opens downward. also, shed some light ... maybe they are very obvious....but I need some sort of clarification... thanks in advance, man.. Check here: http://www.mathopenref.com/quadraticexplorer.htmlhi man thanks a lot I have just visited the site. It is really awesome and very didactic. thanks again, man
|
|
|
Non-Human User
Joined: 09 Sep 2013
Posts: 9894
|
Re: Algebra: Tips and hints
[#permalink]
Show Tags
10 Oct 2018, 10:39
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources
|
|
|
|
Re: Algebra: Tips and hints
[#permalink]
10 Oct 2018, 10:39
|
|
|
|
|
|
|