Question

Using quadratic formula -2x^2+x+8=0

Answer 1

Step by step solution :
Step 1 :

Simplify \[ x-2x^2+8\]

Answer 2

Trying to factor by splitting the middle term

1.1 Factoring \[-2x^2+x+8 \]

Answer 3

The first term is, -2x^2 its coefficient is -2 .
The middle term is, +x its coefficient is 1 .
The last term, "the constant", is +8

Answer 4

Step-1 : Multiply the coefficient of the first term by the constant -2 • 8 = -16

Step-2 : Find two factors of -16 whose sum equals the coefficient of the middle term, which is 1 .

Answer 5

are you understanding this?

Answer 6

-16 + 1 = -15
-8 + 2 = -6
-4 + 4 = 0
-2 + 8 = 6
-1 + 16 = 15


Observation : No two such factors can be found!!
Conclusion : Trinomial can not be factored

Answer 7

Equation at the end of step 1 :

\[ -2x^2+x+8=0 \]

Answer 8

Step 2 :

Solve -2x^2+x+8 = 0

Answer 9

Solve Quadratic Equation by Completing The Square

Solving -2x^2+x+8 = 0 by Completing The Square .

Multiply both sides of the equation by (-1) to obtain positive coefficient for the first term:
2x^2-x-8 = 0 Divide both sides of the equation by 2 to have 1 as the coefficient of the first term :
x^2-(1/2)x-4 = 0

Answer 10

Add 4 to both side of the equation :
x^2-(1/2)x = 4

Now the clever bit: Take the coefficient of x , which is 1/2 , divide by two, giving 1/4 , and finally square it giving 1/16

Add 1/16 to both sides of the equation :
On the right hand side we have :
4 + 1/16 or, (4/1)+(1/16)
The common denominator of the two fractions is 16 Adding (64/16)+(1/16) gives 65/16
So adding to both sides we finally get :
x^2-(1/2)x+(1/16) = 65/16

Adding 1/16 has completed the left hand side into a perfect square :
x^2-(1/2)x+(1/16) =
(x-(1/4)) • (x-(1/4)) =
(x-(1/4))2
Things which are equal to the same thing are also equal to one another. Since
x^2-(1/2)x+(1/16) = 65/16 and
x^2-(1/2)x+(1/16) = (x-(1/4))2
then, according to the law of transitivity,
(x-(1/4))2 = 65/16

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

Now, applying the Square Root Principle to Eq. #2.2.1 we get:
x-(1/4) = √ 65/16

Add 1/4 to both sides to obtain:
x = 1/4 + √ 65/16

Since a square root has two values, one positive and the other negative
x^2 - (1/2)x - 4 = 0
has two solutions:
x = 1/4 + √ 65/16
or
x = 1/4 - √ 65/16

Note that √ 65/16 can be written as
√ 65 / √ 16 which is √ 65 / 4
Solve Quadratic Equation using the Quadratic Formula

2.3 Solving -2x2+x+8 = 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 = 1
C = 8

Accordingly, B2 - 4AC =
1 - (-64) =
65

Applying the quadratic formula :

-1 ± √ 65
x = —————
-4

√ 65 , rounded to 4 decimal digits, is 8.0623
So now we are looking at:
x = ( -1 ± 8.062 ) / -4

Two real solutions:

x =(-1+√65)/-4=-1.766

or:

x =(-1-√65)/-4= 2.266

Two solutions were found :

x =(-1-√65)/-4= 2.266
x =(-1+√65)/-4=-1.766