Quadratic equation refers to polynomial equation that have a general form of ax2+bx+c=0, where a, b and c are co-efficient. a≠0 otherwise it would be a linear equation and c is constant. The quadratic formula is defined as x= (-b±√b

2

-4ac)/2a, where “a” is the co-efficient of the x

2

, b is the linear co-efficient of the x and c is the constant term. Therefore, the quadratic formula involves substituting the co-efficient from a given quadratic equation into the formula. For instance, assume you are given the following equation: 3x

2

+x-2=0. In this case a=3, b=1 and c=-2, substituting this values into the formula we get: x= (-1±√1

2

-4*3*-2)/2*3

Therefore x= (-1±√1+24)/6

Hence x= (-1±√ 25)/6, x= (-1±5)/6. This implies x= (-1+5)/6 or x= (-1-5)/6

Finally x=-2/3 or -1.

Note that the quadratic formula is similar to completing the square method of solving the quadratic equations.

PARABOLA:

Parabola expresses the mathematical curve that an object thrown in the air takes. They take a general form of y=ax2+bx+c. before graphing, there are various factors that must be considered. These include establishing whether the curve opens downward or upward. If a>0 or positive, then curve opens upwards and has a minimum point. Moreover, if a<0 or negative then the curve opens downwards and has a maximum point.

Secondly, the x and y co-ordinates must be established. To calculate the x co-ordinates, you apply the formula x=-b/2a. On the other hand, the y-coordinate is calculated by substituting the value of x found in the above formula in the equation y=ax2+bx+c. Thirdly, the x and y intercepts have to be determined. Y-intercept is established by setting x=0, hence y=c. x-intercept is calculated by setting y=0 thereby obtaining the equation ax

2

+bx+c=0 and solving it to obtain the intercept. If the x-intercept is hard to obtain then find additional points.

On the other hand, to solve the quadratic equations one can apply the factoring method. For instance, assume you are given the following equation: 3×2+4x+1=0. Factoring method involves finding the factors of the 3 that add up to 4, since 3 can be expressed as the product of 3 and 1, at the same time adding the two factors gives the value of 4. Therefore, we will apply 3 and 1 as our factors. Hence, we can write our equation as follows 3×2+3x+1x+1=0, similar to 3x(x+1)+1(x+1)=0. Simplifying the equation, we obtain (3x+1) (x+1) =0

Therefore x=-1 or -1/3.

These inequalities take a general form of ax2 +bx+c≥∕≤0. For example x2-2x-3<0, using the factoring method, the inequality is solved as follows: the first step involves changing the inequality sign to an equal sign x2-2x-3=0.

Factoring x2-2x-3=0 becomes x2-3x+x-3=0

Then simplified as follows x(x-3) +1(x-3) =0

(x+1)(x-3)=0

X=-1 or 3.

On the other hand, a rational inequality is an equation with an unequal sign where the two expressions are rational i.e. polynomials or ratios of polynomials. For instance, assuming you are given the following inequality to solve 3x+4>10. Firstly, you subtract 4 on both sides of the inequality to obtain 3x>6, followed by diving both sides by 3. We obtain x>2, this implies that the solution is set for all numbers greater than 2 or on the interval (2, 9). Note that by dividing on both sides of the inequality by a negative number, the inequality sign has to change.

The advantages of the factoring method include it is simple to use and easier to apply as it only requires one to understand the concept of difference of perfect squares. However, its disadvantage include that it is only used to obtain solutions to equations with a linear term otherwise, it is void. The advantage of the quadratic formula is that it obtains solutions to all quadratic equations with high level of accuracy hence effective. On the hand, it is complicated as it is hard to obtain square root of a large number without a calculator.