# Quadratic equations

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This solution deals with quadratic equations.

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- Quadratic equations

## Step by Step Solution

## Step by step solution :

## Step 1 :

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

1.1 Factoring b^{2}-2b+48

The first term is, b^{2} its coefficient is 1 .

The middle term is, -2b its coefficient is -2 .

The last term, "the constant", is +48

Step-1 : Multiply the coefficient of the first term by the constant 1 • 48 = 48

Step-2 : Find two factors of 48 whose sum equals the coefficient of the middle term, which is -2 .

-48 | + | -1 | = | -49 | ||

-24 | + | -2 | = | -26 | ||

-16 | + | -3 | = | -19 | ||

-12 | + | -4 | = | -16 | ||

-8 | + | -6 | = | -14 | ||

-6 | + | -8 | = | -14 |

For tidiness, printing of 14 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

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

` b`^{2} - 2b + 48 = 0

## Step 2 :

#### Parabola, Finding the Vertex :

2.1 Find the Vertex of y = b^{2}-2b+48

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, 1 , 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,Ab^{2}+Bb+C,the b -coordinate of the vertex is given by -B/(2A) . In our case the b coordinate is 1.0000

Plugging into the parabola formula 1.0000 for b we can calculate the y -coordinate :

y = 1.0 * 1.00 * 1.00 - 2.0 * 1.00 + 48.0

or y = 47.000

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

Root plot for : y = b^{2}-2b+48

Axis of Symmetry (dashed) {b}={ 1.00}

Vertex at {b,y} = { 1.00,47.00}

Function has no real roots

#### Solve Quadratic Equation by Completing The Square

2.2 Solving b^{2}-2b+48 = 0 by Completing The Square .

Subtract 48 from both side of the equation :

b^{2}-2b = -48

Now the clever bit: Take the coefficient of b , which is 2 , divide by two, giving 1 , and finally square it giving 1

Add 1 to both sides of the equation :

On the right hand side we have :

-48 + 1 or, (-48/1)+(1/1)

The common denominator of the two fractions is 1 Adding (-48/1)+(1/1) gives -47/1

So adding to both sides we finally get :

b^{2}-2b+1 = -47

Adding 1 has completed the left hand side into a perfect square :

b^{2}-2b+1 =

(b-1) • (b-1) =

(b-1)^{2}

Things which are equal to the same thing are also equal to one another. Since

b^{2}-2b+1 = -47 and

b^{2}-2b+1 = (b-1)^{2}

then, according to the law of transitivity,

(b-1)^{2} = -47

We'll refer to this Equation as Eq. #2.2.1

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

Note that the square root of

(b-1)^{2} is

(b-1)^{2/2} =

(b-1)^{1} =

b-1

Now, applying the Square Root Principle to Eq. #2.2.1 we get:

b-1 = √ -47

Add 1 to both sides to obtain:

b = 1 + √ -47

In Math, i is called the imaginary unit. It satisfies i^{2} =-1. Both i and -i are the square roots of -1

Since a square root has two values, one positive and the other negative

b^{2} - 2b + 48 = 0

has two solutions:

b = 1 + √ 47 • i

or

b = 1 - √ 47 • i

### Solve Quadratic Equation using the Quadratic Formula

2.3 Solving b^{2}-2b+48 = 0 by the Quadratic Formula .

According to the Quadratic Formula, b , the solution for Ab^{2}+Bb+C = 0 , where A, B and C are numbers, often called coefficients, is given by :

__ __

- B ± √ B^{2}-4AC

b = ————————

2A

In our case, A = 1

B = -2

C = 48

Accordingly, B^{2} - 4AC =

4 - 192 =

-188

Applying the quadratic formula :

2 ± √ -188

b = ——————

2

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)

Both i and -i are the square roots of minus 1

Accordingly,√ -188 =

√ 188 • (-1) =

√ 188 • √ -1 =

± √ 188 • i

Can √ 188 be simplified ?

Yes! The prime factorization of 188 is

2•2•47

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).

√ 188 = √ 2•2•47 =

± 2 • √ 47

√ 47 , rounded to 4 decimal digits, is 6.8557

So now we are looking at:

b = ( 2 ± 2 • 6.856 i ) / 2

Two imaginary solutions :

b =(2+√-188)/2=1+i√ 47 = 1.0000+6.8557i or:

b =(2-√-188)/2=1-i√ 47 = 1.0000-6.8557i

## Two solutions were found :

- b =(2-√-188)/2=1-i√ 47 = 1.0000-6.8557i
- b =(2+√-188)/2=1+i√ 47 = 1.0000+6.8557i