solve x^2 - 19x +14 = -70?

xˆ2 - 19x + 14 = -70

xˆ2 - 19 + 84 = 0

(x - 12)(x - 7) = 0

x = 12
x = 7

Hope I helped!

Move everything to the left side:

x^2 - 19x + 84 = 0

This can be factored:

(x - 12)(x - 7) = 0

x - 12 = 0 or x - 7 = 0

x = 12 or x = 7

x^2 - 19x + 14 = - 70
x^2 - 19x + 14 + 70 = 0
x^2 - 19x + 84 = 0
(x - 12)(x - 7) = 0
x = 12 or 7

*βω*

x^2 - 19x + 84 = 0
(x - 12)(x-7) = 0
x-12 = 0
x=12
or
x-7 = 0
x=7

Question Number 1 :
For this equation x^2 - 19*x + 14 = - 70 , answer the following questions :
A. Find the roots using Quadratic Formula !
B. Use factorization to find the root of the equation !
C. Use completing the square to find the root of the equation !

Answer Number 1 :
First, we have to turn equation : x^2 - 19*x + 14 = - 70 , into a*x^2+b*x+c=0 form.
x^2 - 19*x + 14 = - 70 , move everything in the right hand side, to the left hand side of the equation
<=> x^2 - 19*x + 14 - ( - 70 ) = 0 , which is the same with
<=> x^2 - 19*x + 14 + ( 70 ) =0 , now open the bracket and we get
<=> x^2 - 19*x + 84 = 0

The equation x^2 - 19*x + 84 = 0 is already in a*x^2+b*x+c=0 form.
So we can imply that the value of a = 1, b = -19, c = 84.

1A. Find the roots using Quadratic Formula !
By using abc formula the value of x is both
x1 = (-b+sqrt(b^2-4*a*c))/(2*a) and x2 = (-b-sqrt(b^2-4*a*c))/(2*a)
Since a = 1, b = -19 and c = 84,
then the value a,b and c in the abc formula, can be subtituted.
Which produce x1 = (-(-19) + sqrt( (-19)^2 - 4 * (1)*(84)))/(2*1) and x2 = (-(-19) - sqrt( (-19)^2 - 4 * (1)*(84)))/(2*1)
Which make x1 = ( 19 + sqrt( 361-336))/(2) and x2 = ( 19 - sqrt( 361-336))/(2)
Which is the same with x1 = ( 19 + sqrt( 25))/(2) and x2 = ( 19 - sqrt( 25))/(2)
It imply that x1 = ( 19 + 5 )/(2) and x2 = ( 19 - 5 )/(2)
So we have the answers x1 = 12 and x2 = 7

1B. Use factorization to find the root of the equation !
x^2 - 19*x + 84 = 0
( x - 12 ) * ( x - 7 ) = 0
We get following answers x1 = 12 and x2 = 7

1C. Use completing the square to find the root of the equation !
x^2 - 19*x + 84 = 0 ,divide both side with 1
Then we get x^2 - 19*x + 84 = 0 ,
Which means that the coefficient of x is -19
We have to use the fact that ( x + q )^2 = x^2 + 2*q*x + q^2 , and assume that q = -19/2 = -9.5
By using that fact we turn the equation into x^2 - 19*x + 90.25 - 6.25 = 0
And it is the same with ( x - 9.5 )^2 - 6.25 = 0
And it is the same with (( x - 9.5 ) - 2.5 ) * (( x - 9.5 ) + 2.5 ) = 0
By using the associative law we get ( x - 9.5 - 2.5 ) * ( x - 9.5 + 2.5 ) = 0
Do the addition/subtraction, and we get ( x - 12 ) * ( x - 7 ) = 0
We get following answers x1 = 12 and x2 = 7

x² - 19x + 84 = 0
(x - 12)(x - 7) = 0
x = 12 , x = 7

x^2 - 19x + 14 = -70

1. Add 70 to both sides.
This will cancel out the -70 and change the problem to:
x^2 -19x + 84 = 0

2. Find two numbers that will:
a) equal to the middle number (-19) when you add them together, and
b) equal to the third number (84) when you multiply them together.

In this case the two numbers would be: (-12 and -7)

3. Use each of these numbers to set the equation to zero:
a) (x-12) = 0
b) (x-7) = 0

4. Solve for x on both equations:
a) x =12
b) x = 7

5. Check your answers by plugging in both numbers into the equation:

x^2 - 19x + 14 = -70

(12)^2 - 19(12) + 14 = -70
144 - 228 + 14 = -70

and

(7)^2 - 19 (7) + 14 = -70

x^2 - 19x + 14 = -70
x^2 - 19x + 14 + 70 = 0
x^2 - 19x + 84 = 0
(x - 12)(x - 7) = 0

x - 12 = 0
x = 12

x - 7 = 0
x = 7

∴ x = 12 , 7