# Mathway Без заголовка  24,896,473 solved | 244 online

# 2x2-9x-35=0

## Two solutions were found :

1.  x = -5/2 = -2.500
2.  x = 7

## Step  1  :

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

` (2x2 - 9x) - 35 = 0 `

## Step  2  :

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

2.1      Factoring  2x2-9x-35

The first term is,  2x2  its coefficient is  2 .
The middle term is,  -9x  its coefficient is  -9 .
The last term, “the constant”, is  -35

Step-1 : Multiply the coefficient of the first term by the constant   2 • -35 = -70

Step-2 : Find two factors of  -70  whose sum equals the coefficient of the middle term, which is   -9 .

 -70 + 1 = -69 -35 + 2 = -33 -14 + 5 = -9 That’s it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -14  and  5
2x2 – 14x + 5x – 35

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

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

` (x - 7) • (2x + 5) = 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  :    x-7 = 0

Add  7  to both sides of the equation :

x = 7

#### Solving a Single Variable Equation :

3.3       Solve  :    2x+5 = 0

Subtract  5  from both sides of the equation :

2x = -5

Divide both sides of the equation by 2:
x = -5/2 = -2.500

### Supplement : Solving Quadratic Equation Directly

`Solving  2x2-9x-35  = 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 = 2x2-9x-35

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, 2 , 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   2.2500

Plugging into the parabola formula   2.2500  for  x  we can calculate the  y -coordinate :

y = 2.0 * 2.25 * 2.25 – 9.0 * 2.25 – 35.0
or   y = -45.125

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

Root plot for :  y = 2x2-9x-35
Axis of Symmetry (dashed)  x= 2.25
Vertex at  x,y = 2.25,-45.13
x -Intercepts (Roots) :
Root 1 at  x,y = -2.50, 0.00
Root 2 at  x,y = 7.00, 0.00

#### Solve Quadratic Equation by Completing The Square

4.2      Solving   2x2-9x-35 = 0 by Completing The Square .

Divide both sides of the equation by  2  to have 1 as the coefficient of the first term :
x2-(9/2)x-(35/2) = 0

Add  35/2  to both side of the equation :
x2-(9/2)x = 35/2

Now the clever bit: Take the coefficient of  x , which is  9/2 , divide by two, giving  9/4 , and finally square it giving  81/16

Add  81/16  to both sides of the equation :
On the right hand side we have :
35/2  +  81/16   The common denominator of the two fractions is  16   Adding  (280/16)+(81/16)  gives  361/16
So adding to both sides we finally get :
x2-(9/2)x+(81/16) = 361/16

Adding  81/16  has completed the left hand side into a perfect square :
x2-(9/2)x+(81/16)  =
(x-(9/4)) • (x-(9/4))  =
(x-(9/4))2
Things which are equal to the same thing are also equal to one another. Since
x2-(9/2)x+(81/16) = 361/16 and
x2-(9/2)x+(81/16) = (x-(9/4))2
then, according to the law of transitivity,
(x-(9/4))2 = 361/16

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-(9/4))2   is
(x-(9/4))2/2 =
(x-(9/4))1 =
x-(9/4)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
x-(9/4) = 361/16

Add  9/4  to both sides to obtain:
x = 9/4 + √ 361/16

Since a square root has two values, one positive and the other negative
x2 – (9/2)x – (35/2) = 0
has two solutions:
x = 9/4 + √ 361/16
or
x = 9/4 – √ 361/16

Note that  √ 361/16 can be written as
361  / √ 16   which is 19 / 4

### Solve Quadratic Equation using the Quadratic Formula

4.3      Solving    2x2-9x-35 = 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   =     2
B   =    -9
C   =  -35

Accordingly,  B2  –  4AC   =
81 – (-280) =
361

Applying the quadratic formula :

9 ± √ 361
x  =    —————
4

Can  √ 361 be simplified ?

Yes!   The prime factorization of  361   is
19•19
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).

361   =  √ 19•19   =
±  19 • √ 1   =
±  19

So now we are looking at:
x  =  ( 9 ± 19) / 4

Two real solutions:

x =(9+√361)/4=(9+19)/4= 7.000

or:

x =(9-√361)/4=(9-19)/4= -2.500

## Two solutions were found :

1.  x = -5/2 = -2.500
2.  x = 7

Processing ends successfully

#### Related Links

Mathway
Visit Mathway on the web
Download free on Google Play
Download free on iTunes
Download free on Amazon
Download free in Windows Store

Enter a problem…

Upgrade
About
Help

Sign In
Sign Up
Hope that helps!
You’re welcome!
Let me take a look…
You’ll be able to enter math problems once our session is over.
Upgrade
About
Help

Sign In
Sign Up

# Algebra Examples

Popular Problems
Algebra
Factor 2x^2-9x-35

Factor by grouping.

Tap for more steps…

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Tap for more steps…

Factor out of .
Rewrite as plus
Apply the distributive property.
Remove parentheses.
Factor out the greatest common factor from each group.

Tap for more steps…

Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .

Mathway requires javascript and a modern browser.