## Quadratic equations

### Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x2" was replaced by "x^2".

### Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

x^2-15*x-(2)=0

### Step by step solution :

### Step 1 :

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

1.1 Factoring x^{2}-15x-2

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

The middle term is, -15x its coefficient is -15 .

The last term, "the constant", is -2

Step-1 : Multiply the coefficient of the first term by the constant 1 • -2 = -2

Step-2 : Find two factors of -2 whose sum equals the coefficient of the middle term, which is -15 .

-2 | + | 1 | = | -1 | |

-1 | + | 2 | = | 1 |

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

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

x^{2}- 15x - 2 = 0

### Step 2 :

#### Parabola, Finding the Vertex :

2.1 Find the Vertex of y = x^{2}-15x-2

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,Ax^{2}+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 7.5000

Plugging into the parabola formula 7.5000 for x we can calculate the y -coordinate :

y = 1.0 * 7.50 * 7.50 - 15.0 * 7.50 - 2.0

or y = -58.250

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

Root plot for : y = x^{2}-15x-2

Axis of Symmetry (dashed) {x}={ 7.50}

Vertex at {x,y} = { 7.50,-58.25}

x -Intercepts (Roots) :

Root 1 at {x,y} = {-0.13, 0.00}

Root 2 at {x,y} = {15.13, 0.00}

#### Solve Quadratic Equation by Completing The Square

2.2 Solving x^{2}-15x-2 = 0 by Completing The Square .

Add 2 to both side of the equation :

x^{2}-15x = 2

Now the clever bit: Take the coefficient of x , which is 15 , divide by two, giving 15/2 , and finally square it giving 225/4

Add 225/4 to both sides of the equation :

On the right hand side we have :

2 + 225/4 or, (2/1)+(225/4)

The common denominator of the two fractions is 4 Adding (8/4)+(225/4) gives 233/4

So adding to both sides we finally get :

x^{2}-15x+(225/4) = 233/4

Adding 225/4 has completed the left hand side into a perfect square :

x^{2}-15x+(225/4) =

(x-(15/2)) • (x-(15/2)) =

(x-(15/2))^{2}

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

x^{2}-15x+(225/4) = 233/4 and

x^{2}-15x+(225/4) = (x-(15/2))^{2}

then, according to the law of transitivity,

(x-(15/2))^{2} = 233/4

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

(x-(15/2))^{2} is

(x-(15/2))^{2/2} =

(x-(15/2))^{1} =

x-(15/2)

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

x-(15/2) = √ 233/4

Add 15/2 to both sides to obtain:

x = 15/2 + √ 233/4

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

x^{2} - 15x - 2 = 0

has two solutions:

x = 15/2 + √ 233/4

or

x = 15/2 - √ 233/4

Note that √ 233/4 can be written as

√ 233 / √ 4 which is √ 233 / 2

### Solve Quadratic Equation using the Quadratic Formula

2.3 Solving x^{2}-15x-2 = 0 by the Quadratic Formula .

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

__ __

- B ± √ B^{2}-4AC

x = ————————

2A

In our case, A = 1

B = -15

C = -2

Accordingly, B^{2} - 4AC =

225 - (-8) =

233

Applying the quadratic formula :

15 ± √ 233

x = ——————

2

√ 233 , rounded to 4 decimal digits, is 15.2643

So now we are looking at:

x = ( 15 ± 15.264 ) / 2

Two real solutions:

x =(15+√233)/2=15.132

or:

x =(15-√233)/2=-0.132

### Two solutions were found :

- x =(15-√233)/2=-0.132
- x =(15+√233)/2=15.132

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## Quadratic equations

### Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x2" was replaced by "x^2".

### Step by step solution :

### Step 1 :

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

1.1 Factoring x^{2}-15x-500

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

The middle term is, -15x its coefficient is -15 .

The last term, "the constant", is -500

Step-1 : Multiply the coefficient of the first term by the constant 1 • -500 = -500

Step-2 : Find two factors of -500 whose sum equals the coefficient of the middle term, which is -15 .

-500 | + | 1 | = | -499 | |

-250 | + | 2 | = | -248 | |

-125 | + | 4 | = | -121 | |

-100 | + | 5 | = | -95 | |

-50 | + | 10 | = | -40 | |

-25 | + | 20 | = | -5 | |

-20 | + | 25 | = | 5 | |

-10 | + | 50 | = | 40 | |

-5 | + | 100 | = | 95 | |

-4 | + | 125 | = | 121 | |

-2 | + | 250 | = | 248 | |

-1 | + | 500 | = | 499 |

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

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

x^{2}- 15x - 500 = 0

### Step 2 :

#### Parabola, Finding the Vertex :

2.1 Find the Vertex of y = x^{2}-15x-500

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,Ax^{2}+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 7.5000

Plugging into the parabola formula 7.5000 for x we can calculate the y -coordinate :

y = 1.0 * 7.50 * 7.50 - 15.0 * 7.50 - 500.0

or y = -556.250

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

Root plot for : y = x^{2}-15x-500

Axis of Symmetry (dashed) {x}={ 7.50}

Vertex at {x,y} = { 7.50,-556.25}

x -Intercepts (Roots) :

Root 1 at {x,y} = {-16.08, 0.00}

Root 2 at {x,y} = {31.08, 0.00}

#### Solve Quadratic Equation by Completing The Square

2.2 Solving x^{2}-15x-500 = 0 by Completing The Square .

Add 500 to both side of the equation :

x^{2}-15x = 500

Now the clever bit: Take the coefficient of x , which is 15 , divide by two, giving 15/2 , and finally square it giving 225/4

Add 225/4 to both sides of the equation :

On the right hand side we have :

500 + 225/4 or, (500/1)+(225/4)

The common denominator of the two fractions is 4 Adding (2000/4)+(225/4) gives 2225/4

So adding to both sides we finally get :

x^{2}-15x+(225/4) = 2225/4

Adding 225/4 has completed the left hand side into a perfect square :

x^{2}-15x+(225/4) =

(x-(15/2)) • (x-(15/2)) =

(x-(15/2))^{2}

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

x^{2}-15x+(225/4) = 2225/4 and

x^{2}-15x+(225/4) = (x-(15/2))^{2}

then, according to the law of transitivity,

(x-(15/2))^{2} = 2225/4

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

(x-(15/2))^{2} is

(x-(15/2))^{2/2} =

(x-(15/2))^{1} =

x-(15/2)

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

x-(15/2) = √ 2225/4

Add 15/2 to both sides to obtain:

x = 15/2 + √ 2225/4

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

x^{2} - 15x - 500 = 0

has two solutions:

x = 15/2 + √ 2225/4

or

x = 15/2 - √ 2225/4

Note that √ 2225/4 can be written as

√ 2225 / √ 4 which is √ 2225 / 2

### Solve Quadratic Equation using the Quadratic Formula

2.3 Solving x^{2}-15x-500 = 0 by the Quadratic Formula .

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

__ __

- B ± √ B^{2}-4AC

x = ————————

2A

In our case, A = 1

B = -15

C = -500

Accordingly, B^{2} - 4AC =

225 - (-2000) =

2225

Applying the quadratic formula :

15 ± √ 2225

x = ——————

2

Can √ 2225 be simplified ?

Yes! The prime factorization of 2225 is

5•5•89

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).

√ 2225 = √ 5•5•89 =

± 5 • √ 89

√ 89 , rounded to 4 decimal digits, is 9.4340

So now we are looking at:

x = ( 15 ± 5 • 9.434 ) / 2

Two real solutions:

x =(15+√2225)/2=(15+5√ 89 )/2= 31.085

or:

x =(15-√2225)/2=(15-5√ 89 )/2= -16.085

### Two solutions were found :

- x =(15-√2225)/2=(15-5√ 89 )/2= -16.085
- x =(15+√2225)/2=(15+5√ 89 )/2= 31.085

## 15x x^2

### Simple and best practice solution for x^2-15x-2=0 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so don`t hesitate to use it as a solution of your homework.

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### Solution for x^2-15x-2=0 equation:

Simplifying x

^{2}+ -15x + -2 = 0 Reorder the terms: -2 + -15x + x

^{2}= 0 Solving -2 + -15x + x

^{2}= 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + -15x + 2 + x

^{2}= 0 + 2 Reorder the terms: -2 + 2 + -15x + x

^{2}= 0 + 2 Combine like terms: -2 + 2 = 0 0 + -15x + x

^{2}= 0 + 2 -15x + x

^{2}= 0 + 2 Combine like terms: 0 + 2 = 2 -15x + x

^{2}= 2 The x term is -15x. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15x + 56.25 + x

^{2}= 2 + 56.25 Reorder the terms: 56.25 + -15x + x

^{2}= 2 + 56.25 Combine like terms: 2 + 56.25 = 58.25 56.25 + -15x + x

^{2}= 58.25 Factor a perfect square on the left side: (x + -7.5)(x + -7.5) = 58.25 Calculate the square root of the right side: 7.632168761 Break this problem into two subproblems by setting (x + -7.5) equal to 7.632168761 and -7.632168761.

#### Subproblem 1

x + -7.5 = 7.632168761 Simplifying x + -7.5 = 7.632168761 Reorder the terms: -7.5 + x = 7.632168761 Solving -7.5 + x = 7.632168761 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = 7.632168761 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = 7.632168761 + 7.5 x = 7.632168761 + 7.5 Combine like terms: 7.632168761 + 7.5 = 15.132168761 x = 15.132168761 Simplifying x = 15.132168761#### Subproblem 2

x + -7.5 = -7.632168761 Simplifying x + -7.5 = -7.632168761 Reorder the terms: -7.5 + x = -7.632168761 Solving -7.5 + x = -7.632168761 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = -7.632168761 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = -7.632168761 + 7.5 x = -7.632168761 + 7.5 Combine like terms: -7.632168761 + 7.5 = -0.132168761 x = -0.132168761 Simplifying x = -0.132168761#### Solution

The solution to the problem is based on the solutions from the subproblems. x = {15.132168761, -0.132168761}#### You can always share this solution

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