Solving Equations Homework Help

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Solving Equations

The linear one variable equation is of the form, ax+b=c where a,b and c are constants. The solution of the equation is satisfied that equation. The linear equations can be simplified by using the below properties:

1. If a=b, then a+c=b+ c.

2. If a=b, then a−c=b− c.

3. If a=b, then ac=bc.

4. If a=b, then

, where


Steps to solve one variable linear equation:

Step 1:

The parentheses involved in the equations are removed by applying the distributive property and the fractions are eliminated by multiplying each side by LCD of that fractions.

Step 2:

Use the properties and retain the unknown variable on one side and shift the remaining terms on the other side.

Step 3:

Perform the elementary calculations to obtain the solution.

Example 1:

Consider the equation



Multiply both sides by 2.


Isolate x variable.


Solving one variable quadratic equations:

A quadratic equation is a single variable polynomial equation of order 2 in the form ax2+bx+c=0 with
where a,b,c are constants and x is unknown variable.

Using factor method or quadratic formula to solving this equation.

Factoring a quadratic equation:

Step 1: Consider the equation ax²+bx+ c=0

Step 2: Multiply the a term of the co-efficient of x2 with constant c term.

Step 3: Obtain two factors r and s, that multiplies together to get ac and add together to get b.

Step 4: Replace the middle term bx with the factors (r+ s)x.

Step 5: Solve by grouping. That is taking the equation in two groups. Obtain the GCF from each group. Take the similar groups and combine remaining terms.

Step 6: Equate zero to the factors gives the required two solutions (roots).

Example 2:

Consider the polynomial 3x²+2x–8=0.

The product of 3 and − 8; is −24. Thus, obtain factors such that their product is - 24 and the sum is 2. Therefore, the factors are 6 and−4.


Equating the factors to zero.

(x+2)=0 or (3x−4)=0

Thus, the solutions are x=–2and
.

Solving quadratic equation by using formula:

The quadratic formula is
.

Example 3:

Consider a quadratic equation x²+2x+5=0 . Since the given quadratic equation cannot be factored easily, the required roots can be obtained by using quadratic formula method as follows:

Here a=1,b=2,and c=5.

Therefore,



Here, the roots are complex. That is, x1=–1+2i and x2=–1–2i.

Solving one variable higher degree equation:

The equation with degree up to 5 can be solved by using the rational root theorem, which helps to obtain one factor and then the remaining expression can be reduced to lower degree equation by using synthetic division or long division method. Continue this process (or apply factor method) until all the roots are obtained.

Suppose the degree of the equation is higher (greater) than 5, then rational root theorem works can be applied only to solve few simple problems. Thus, in order to solve the complex problems, various methods or techniques can be applied say, numerical methods like Newton-Raphson method or Root-finding algorithms can be used to obtain the roots.