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.



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