XXI Quadratics

Graphs of Quadratic Function

y = x^2

This function's graph is a combination of two parallel curves called a parabola. There is one value of yy for each positive and negative pair of value of xx. The graph has a minimum value of 0 at the origin, which is also the turning point.
Due to its nature the graph can be used to find square roots.

y = -x^2

Same as the graph of the above function. However all values for yy are negative. Also the curves are inverted and moving downwards.

y = ax^2

Same as the first function, however aa controls the steepness of the curve. The higher its value, the steeper the curve.

y = x^2 +/- a

Same as the first function but the graph is raised or lowered according to the value of aa

y = (x-a)^2

Displaces the graph aa units to the right, or to the left if its value is added instead of subtracted.

y = (x-b)^2 + a

A general function for any possible position on the coordinate plane with any possible steepness. Implies the standard form of

y=ax2+bx+cy = ax^2 + bx + c

ax^2 + bx + c = 0

Describes the points where the curve cuts the x-axis. They can be read from a plotted graph where yy equals 00.

Quadratic Inequalities

Inequalities such as x24x+3>0x^2-4x+3 > 0 and x24x+3<0x^2-4x+3 < 0 define all the values of xx above and below all points where y=0y = 0. Therefore they can be solved for all points on the parabola that fall into the regions.

On the other hand general inequalities such as y>x25x+4y > x^2-5x+4 and y<x25x+4y < x^2-5x+4 describe the regions above and below the curve y=x25x+4y = x^2-5x+4.

Quadratic Equations

Algebraic techniques can be used to solve such equations with any required degree of accuracy.

For example, an equation such as (x1)24=0(x-1)^2 - 4 = 0 can be put transformed to find the corresponding values of xx where y=0y = 0.

(x1)24=0(x-1)^2 - 4 = 0
=(x1)2=4= (x-1)^2 = 4
=x1=±2= x-1 = \pm 2

x1=2x-1 = 2
x=3x = 3

x1=2x-1 = -2
x=1x = -1

So two valid points of the curve are (3,0)(3,0) and (1,0)(-1,0).

Solving by Completing the Square

Given the equation x22x3=0x^2-2x-3 = 0

  1. Remove the constant on the right:
    x22x=3x^2-2x = 3

  2. Divide throughout by the coefficient of x2x^2 if it is anything but unity

  3. Add a number that produces a complete square on the left side: (Always the square of half the coefficient of xx)

x22x+(1)2=3+1x^2-2x+(-1)^2 = 3 + 1

This factors into: (x1)2=4(x-1)^2=4

Solving by Factorization

If the product of two factors is 00, then at least one of the factors has to be 00 as well.

Therefore factoring an equation can lead to the values of xx where y=0y = 0.

Given (x1)(x3)=0(x-1)(x-3)= 0

x1=0x-1 = 0
x=1x = 1

x3=0x-3 = 0
x=3x = 3

This method can be used to solve equations of higher degrees than x2x^2.

General Formula

Given any quadratic ax2+bx+c=0ax^2+bx+c = 0

x1,2=b±b24ac22ax_{1,2} = \frac{-b\pm\sqrt[2]{b^2-4ac}}{2a}

Simultaneous equations of the second degree

A term of xyxy is a term of the second degree much like x2x^2 or y2y^2.

Therefore <?kt xyz ?> is a term of the third degree.

Simultaneous quadratic equations are only solvable in a very narrow range of cases.

One such case is when one of the equations is a linear equation. In this case the approach to solving is to solve the linear equation for one unknown and substitute it into the quadratic equation.


xy=2y=x2x - y = 2 \Rarr y = x-2
x2+xy=60x2+x(x2)=60x^2+xy = 60 \Rarr x^2+x(x-2)=60
2x22x60=02x^2 - 2x - 60 = 0

If the resulting equation happens to have a negative value for aa, it is correct and flipping signs by multiplying by 1-1 is not legal.

Solving Quadratic Inequalities

Factorization allows to solve these inequalities. Given x26x+8<0x^2-6x+8<0, results in (x2)(x4)<0(x-2)(x-4)<0 when factorized.

Since the product of the factors is negative, one factor has to come out negative while the other has to be positive.

In case of (x2)(x4)>0(x-2)(x-4) > 0 they would both have to yield the same sign for the product to be a positive number.

If x>2x > 2 then (x2)>0(x-2) > 0 On the other hand if x<2x < 2 then (x2)<0(x-2) < 0.

For (x2)(x4)(x-2)(x-4) to yield a pair of opposing signs, xx has to be a value between 22 and 44.

Thus 2<x<42 < x < 4.

The solution for a positive product would be either x<2x < 2 or x>4x > 4, that is a value that yields the same sign for both factors.

This method also works for inequalities of higher degrees.

Number lines can come handy when solving these inequalities.