Question

how to factor 6x^2-9x-10

Answer 1

Step by step solution :

Step 1 :

Simplify 6x2-9x - 10

Answer 2

Either the quadratic formula or complete the square.

Answer 3

Solve Quadratic Equation by Completing The Square

2.2 Solving 6x2-9x-10 = 0 by Completing The Square .

Divide both sides of the equation by 6 to have 1 as the coefficient of the first term :
x2-(3/2)x-(5/3) = 0

Add 5/3 to both side of the equation :
x2-(3/2)x = 5/3

Now the clever bit: Take the coefficient of x , which is 3/2 , divide by two, giving 3/4 , and finally square it giving 9/16

Add 9/16 to both sides of the equation :
On the right hand side we have :
5/3 + 9/16 The common denominator of the two fractions is 48 Adding (80/48)+(27/48) gives 107/48
So adding to both sides we finally get :
x2-(3/2)x+(9/16) = 107/48

Adding 9/16 has completed the left hand side into a perfect square :
x2-(3/2)x+(9/16) =
(x-(3/4)) • (x-(3/4)) =
(x-(3/4))2
Things which are equal to the same thing are also equal to one another. Since
x2-(3/2)x+(9/16) = 107/48 and
x2-(3/2)x+(9/16) = (x-(3/4))2
then, according to the law of transitivity,
(x-(3/4))2 = 107/48

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-(3/4))2 is
(x-(3/4))2/2 =
(x-(3/4))1 =
x-(3/4)

Now, applying the Square Root Principle to Eq. #2.2.1 we get:
x-(3/4) = √ 107/48

Add 3/4 to both sides to obtain:
x = 3/4 + √ 107/48

Since a square root has two values, one positive and the other negative
x2 - (3/2)x - (5/3) = 0
has two solutions:
x = 3/4 + √ 107/48
or
x = 3/4 - √ 107/48

Note that √ 107/48 can be written as
√ 107 / √ 48

Solve Quadratic Equation using the Quadratic Formula

2.3 Solving 6x2-9x-10 = 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 = -9
C = -10

Accordingly, B2 - 4AC =
81 - (-240) =
321

Applying the quadratic formula :

9 ± √ 321
x = —————
12

√ 321 , rounded to 4 decimal digits, is 17.9165
So now we are looking at:
x = ( 9 ± 17.916 ) / 12

Two real solutions:

x =(9+√321)/12=3/4+1/12√ 321 = 2.243

or:

x =(9-√321)/12=3/4-1/12√ 321 = -0.743


Two solutions were found :

x =(9-√321)/12=3/4-1/12√ 321 = -0.743
x =(9+√321)/12=3/4+1/12√ 321 = 2.243