# What is x if 7x 7 98

## Two solutions were found :

1.  x = -14/3 = -4.667
2.  x = 7/2 = 3.500

## Step  1  :

#### Equation at the end of step  1  :

((2•3x2) + 7x) - 98 = 0

## Step  2  :

#### Trying to factor by splitting the middle term

2.1     Factoring  6x2+7x-98

The first term is,  6x2  its coefficient is  6 .
The middle term is,  +7x  its coefficient is  7 .
The last term, "the constant", is  -98

Step-1 : Multiply the coefficient of the first term by the constant   6 • -98 = -588

Step-2 : Find two factors of  -588  whose sum equals the coefficient of the middle term, which is   7 .

 -588 + 1 = -587 -294 + 2 = -292 -196 + 3 = -193 -147 + 4 = -143 -98 + 6 = -92 -84 + 7 = -77 -49 + 12 = -37 -42 + 14 = -28 -28 + 21 = -7 -21 + 28 = 7 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -21  and  28
6x2 - 21x + 28x - 98

Step-4 : Add up the first 2 terms, pulling out like factors :
3x • (2x-7)
Add up the last 2 terms, pulling out common factors :
14 • (2x-7)
Step-5 : Add up the four terms of step 4 :
(3x+14)  •  (2x-7)
Which is the desired factorization

#### Equation at the end of step  2  :

(2x - 7) • (3x + 14) = 0

## Step  3  :

#### Theory - Roots of a product :

3.1    A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

#### Solving a Single Variable Equation :

3.2      Solve  :    2x-7 = 0

Add  7  to both sides of the equation :
2x = 7
Divide both sides of the equation by 2:
x = 7/2 = 3.500

#### Solving a Single Variable Equation :

3.3      Solve  :    3x+14 = 0

Subtract  14  from both sides of the equation :
3x = -14
Divide both sides of the equation by 3:
x = -14/3 = -4.667

### Supplement : Solving Quadratic Equation Directly

Solving  6x2+7x-98  = 0 directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

#### Parabola, Finding the Vertex :

4.1      Find the Vertex of   y = 6x2+7x-98

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 6 , is positive (greater than zero).

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.

For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is  -0.5833

Plugging into the parabola formula  -0.5833  for  x  we can calculate the  y -coordinate :
y = 6.0 * -0.58 * -0.58 + 7.0 * -0.58 - 98.0
or   y = -100.042

#### Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 6x2+7x-98
Axis of Symmetry (dashed)  {x}={-0.58}
Vertex at  {x,y} = {-0.58,-100.04}
x -Intercepts (Roots) :
Root 1 at  {x,y} = {-4.67, 0.00}
Root 2 at  {x,y} = { 3.50, 0.00}

#### Solve Quadratic Equation by Completing The Square

4.2     Solving   6x2+7x-98 = 0 by Completing The Square .

Divide both sides of the equation by  6  to have 1 as the coefficient of the first term :
x2+(7/6)x-(49/3) = 0

Add  49/3  to both side of the equation :
x2+(7/6)x = 49/3

Now the clever bit: Take the coefficient of  x , which is  7/6 , divide by two, giving  7/12 , and finally square it giving  49/144

Add  49/144  to both sides of the equation :
On the right hand side we have :
49/3  +  49/144   The common denominator of the two fractions is  144   Adding  (2352/144)+(49/144)  gives  2401/144
So adding to both sides we finally get :
x2+(7/6)x+(49/144) = 2401/144

Adding  49/144  has completed the left hand side into a perfect square :
x2+(7/6)x+(49/144)  =
(x+(7/12)) • (x+(7/12))  =
(x+(7/12))2
Things which are equal to the same thing are also equal to one another. Since
x2+(7/6)x+(49/144) = 2401/144 and
x2+(7/6)x+(49/144) = (x+(7/12))2
then, according to the law of transitivity,
(x+(7/12))2 = 2401/144

We'll refer to this Equation as  Eq. #4.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
(x+(7/12))2  is
(x+(7/12))2/2 =
(x+(7/12))1 =
x+(7/12)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
x+(7/12) = √ 2401/144

Subtract  7/12  from both sides to obtain:
x = -7/12 + √ 2401/144

Since a square root has two values, one positive and the other negative
x2 + (7/6)x - (49/3) = 0
has two solutions:
x = -7/12 + √ 2401/144
or
x = -7/12 - √ 2401/144

Note that  √ 2401/144 can be written as
√ 2401  / √ 144   which is 49 / 12

4.3     Solving    6x2+7x-98 = 0 by the Quadratic Formula .

According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :

- B  ±  √ B2-4AC
x =   ————————
2A

In our case,  A   =     6
B   =    7
C   =  -98

Accordingly,  B2  -  4AC   =
49 - (-2352) =
2401

-7 ± √ 2401
x  =    ——————
12

Can  √ 2401 be simplified ?

Yes!   The prime factorization of  2401   is
7•7•7•7
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 2401   =  √ 7•7•7•7  =7•7•√ 1   =
±  49 • √ 1   =
±  49

So now we are looking at:
x  =  ( -7 ± 49) / 12

Two real solutions:

x =(-7+√2401)/12=(-7+49)/12= 3.500

or:

x =(-7-√2401)/12=(-7-49)/12= -4.667

## Two solutions were found :

1.  x = -14/3 = -4.667
2.  x = 7/2 = 3.500

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