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Algebra: Tips and hints

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Bunuel
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Algebra: Tips and hints [#permalink] New post   16 Jul 2014, 11:30
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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:

  • 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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Re: Algebra: Tips and hints [#permalink] New post   17 Jul 2014, 14:28
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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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Re: Algebra: Tips and hints [#permalink] New post   20 Sep 2017, 10:15
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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:

  • 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: https://www.mathopenref.com/quadraticexplorer.html
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testcracker
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Algebra: Tips and hints [#permalink] New post   Updated on: 20 Sep 2017, 10:34
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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:

  • 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: https://gmatclub.com/forum/algebra-101576.html

DS Algebra Questions to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=29
PS Algebra Questions to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=50

Special algebra set: https://gmatclub.com/forum/new-algebra-set-149349.html


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, 10:10.
Last edited by testcracker on 20 Sep 2017, 10:34, edited 1 time in total.
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minirana
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Re: Algebra: Tips and hints [#permalink] New post   17 Jul 2020, 22:02
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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:

  • 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: https://www.mathopenref.com/quadraticexplorer.html


Hi Bunuel,

The site was very helpful to understand the different dynamics of quadratic equation and parabola.

I am unable to understand :
[list][*] The larger the absolute value of \(a\), the steeper (or thinner) the parabola is, since the value of y is increased more quickly.

Taking the given example :
This is the graph of the equation y = 2x2+3x+4. Note how it combines the effects of the three terms. Play with various values of a, b and c.

Changing c moves it up and down - It defines the y intercept so i understand why it is moving up and down
Changing b changes the slope - when it is a parabola , which points are they considering for the slope as m = y2-y1/x2-x1
Changing a alters the curvature of the parabolic element - i understand if a is +ve the curve is upwards , if -ve the curve is downwards. But what i dont understand that the larger the value of a why is it getting steeper/ thinner? I thought eg. if a = 2 ; then the parabola considers +2 and -2 on the x-axis and thus should be opposite.

Thanks.
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Re: Algebra: Tips and hints [#permalink] New post   20 Sep 2017, 14:29
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:

  • 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: https://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] New post   31 Dec 2019, 09:44
This is helpful, thanks.
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Re: Algebra: Tips and hints [#permalink] New post   17 Dec 2023, 05:28
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Re: Algebra: Tips and hints [#permalink]
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