how do you solve 8x+14=x^2+1?

Question Number 1 :

For this equation 8*x + 14 = x^2 + 1 , answer the following questions :

A. Find the roots using Quadratic Formula !

B. Use completing the square to find the root of the equation !

Answer Number 1 :

First, we have to turn equation : 8*x + 14 = x^2 + 1 , into a*x^2+b*x+c=0 form.

8*x + 14 = x^2 + 1 , move everything in the right hand side, to the left hand side of the equation

<=> 8*x + 14 - ( x^2 + 1 ) = 0 , which is the same with

<=> 8*x + 14 + ( -x^2 - 1 ) =0 , now open the bracket and we get

<=> -x^2 + 8*x + 13 = 0

The equation -x^2 + 8*x + 13 = 0 is already in a*x^2+b*x+c=0 form.

By matching the constant position, we can derive that the value of a = -1, b = 8, c = 13.

1A. Find the roots using Quadratic Formula !

Use abc formula and you get either

x1 = (-b+sqrt(b^2-4*a*c))/(2*a) or x2 = (-b-sqrt(b^2-4*a*c))/(2*a)

We had know that a = -1, b = 8 and c = 13,

then the value a,b and c in the abc formula, can be subtituted.

Which produce x1 = (-(8) + sqrt( (8)^2 - 4 * (-1)*(13)))/(2*-1) and x2 = (-(8) - sqrt( (8)^2 - 4 * (-1)*(13)))/(2*-1)

Which is the same with x1 = ( -8 + sqrt( 64+52))/(-2) and x2 = ( -8 - sqrt( 64+52))/(-2)

Which make x1 = ( -8 + sqrt( 116))/(-2) and x2 = ( -8 - sqrt( 116))/(-2)

We can get x1 = ( -8 + 10.770329614269 )/(-2) and x2 = ( -8 - 10.770329614269 )/(-2)

So we have the answers x1 = -1.3851648071345 and x2 = 9.3851648071345

1B. Use completing the square to find the root of the equation !

-x^2 + 8*x + 13 = 0 ,divide both side with -1

So we get x^2 - 8*x - 13 = 0 ,

The coefficient of x is -8

We have to use the fact that ( x + q )^2 = x^2 + 2*q*x + q^2 , and assume that q = -8/2 = -4

Next, we have to separate the constant to form x^2 - 8*x + 16 - 29 = 0

So we will get ( x - 4 )^2 - 29 = 0

So we will get (( x - 4 ) - 5.3851648071345 ) * (( x - 4 ) + 5.3851648071345 ) = 0

By using the associative law we get ( x - 4 - 5.3851648071345 ) * ( x - 4 + 5.3851648071345 ) = 0

Do the addition/subtraction, and we get ( x - 9.3851648071345 ) * ( x + 1.3851648071345 ) = 0

So we have the answers x1 = 9.3851648071345 and x2 = -1.3851648071345

you need to change it so that you end up with an equation that looks like this..

ax^2+bx+c=0

so first you need to take the 1 from the right side of the equation and take it from 14.

so now you have 8x+13=x^2

Now you need to take x^2 from each side..

-x^2+8x+13=0

now, because this equation can't be put into 2 brackets, you need to use the quadratic formula.

where

a = the number infront of x^2

b = the number in front of x

c = the number at the end.

(look up the quadratic formula, but from there on its not too hard.)

isolate 0 by transposing every term in one side of equation:

x^2 - 8x - 13 = 0

find the roots of x, x is unfactorable so use the quadratic equation.

quadratic equation:

let ax^2 + bx + c = 0

then,

x = {-b + sqrt(b^2 - 4ac)}/2a and {-b - sqrt(b^2 - 4ac)}/2a