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Algebra: Tips and hints

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Algebra: Tips and hints  [#permalink]

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New post 16 Jul 2014, 10:30
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48

Algebra: Tips and hints



!
This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

Algebraic Identities
1. \((x+y)^2=x^2+y^2+2xy\)
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. \(x^3-y^3=(x-y)(x^2+y^2+xy)\)

Quadratics

The 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:
Image
  • 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 theorem

Viete'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 avoid
Never 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 \(x=1\).

This week's PS question
This week's DS Question

Theory on Algebra: algebra-101576.html

DS Algebra Questions to practice: search.php?search_id=tag&tag_id=29
PS Algebra Questions to practice: search.php?search_id=tag&tag_id=50

Special algebra set: new-algebra-set-149349.html


Please share your Algebra tips below and get kudos point. Thank you.
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Resources:
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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.


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Re: Algebra: Tips and hints  [#permalink]

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New post 17 Jul 2014, 13:28
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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.html

but sometimes it will be hidden, as in PS 199 in OG 13: topic-137149.html

In 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.
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Algebra: Tips and hints  [#permalink]

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New post Updated on: 20 Sep 2017, 09:34
Bunuel wrote:

Algebra: Tips and hints



!
This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

Algebraic Identities
1. \((x+y)^2=x^2+y^2+2xy\)
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. \(x^3-y^3=(x-y)(x^2+y^2+xy)\)

Quadratics

The 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:
Image
  • 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 theorem

Viete'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 avoid
Never 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.
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Joined: 02 Sep 2009
Posts: 53067
Re: Algebra: Tips and hints  [#permalink]

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New post 20 Sep 2017, 09:15
gmatcracker2017 wrote:
Bunuel wrote:

Algebra: Tips and hints



!
This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

Algebraic Identities
1. \((x+y)^2=x^2+y^2+2xy\)
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. \(x^3-y^3=(x-y)(x^2+y^2+xy)\)

Quadratics

The 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:
Image
  • 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 theorem

Viete'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 avoid
Never 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
Senior Manager
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Status: love the club...
Joined: 24 Mar 2015
Posts: 273
Re: Algebra: Tips and hints  [#permalink]

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New post 20 Sep 2017, 13:29
Bunuel wrote:
gmatcracker2017 wrote:
Bunuel wrote:

Algebra: Tips and hints



!
This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

Algebraic Identities
1. \((x+y)^2=x^2+y^2+2xy\)
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. \(x^3-y^3=(x-y)(x^2+y^2+xy)\)

Quadratics

The 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:
Image
  • 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 theorem

Viete'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 avoid
Never 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



hi man
thanks a lot :grin:

I have just visited the site. It is really awesome and very didactic.
thanks again, man 8-)
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Re: Algebra: Tips and hints  [#permalink]

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New post 10 Oct 2018, 10:39
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Re: Algebra: Tips and hints   [#permalink] 10 Oct 2018, 10:39
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