Solve the equation -x^2+4=2x^2-5?

-x^2 + 4 = 2x^2 - 5
-x^2 - 2x^2 = -4 - 5
-3x^2 = -9
x^2 = -9/(-3)
x^2 = 3
x = ±√3

3 x ² = 9

x ² = 3

x = ± √3

1) x^2 + 3x - 10 = 0 What two numbers multiply to make -10 and add to make 3? The answer is +5 and -2. That's how it factors. (x + 5)(x - 2) = 0, which means x = {-5, 2} 3) 2x^2 - 3x - 2 = 0 Multiply the coefficient of the first term and the last term. Then try to get factors of that product to add up to the middle term. 2 times (-2) is equal to -4. What two factors of -4 add up to -3? The answer is -4 and +1. Split -3x into -4x and -x. 2x^2 - 4x + x - 2 = 0 Group this; factor the first two terms and the last two terms. 2x(x - 2) + (x - 2) = 0 Group (x - 2) out of this entire equation. (x - 2)(2x + 1) = 0 Equate each factor to 0. x - 2 = 0 2x + 1 = 0 x = 2 2x = -1, x = -1/2 x = {2, -1/2} 5) x^2 + 6x + 5 = 0 What two numbers multiply to make 5 and add to make 6? The answer is 5 and 1. (x + 5)(x + 1) = 0 x = {-5, -1}

Question Number 1 :
For this equation -x^2 + 4 = 2*x^2 - 5 , 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 : -x^2 + 4 = 2*x^2 - 5 , into a*x^2+b*x+c=0 form.
-x^2 + 4 = 2*x^2 - 5 , move everything in the right hand side, to the left hand side of the equation
<=> -x^2 + 4 - ( 2*x^2 - 5 ) = 0 , which is the same with
<=> -x^2 + 4 + ( -2*x^2 + 5 ) =0 , now open the bracket and we get
<=> -3*x^2 + 9 = 0

The equation -3*x^2 + 9 = 0 is already in a*x^2+b*x+c=0 form.
So we can imply that the value of a = -3, b = 0, c = 9.

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)
As a = -3, b = 0 and c = 9,
we need to subtitute a,b,c in the abc formula, with thos values.
So we get x1 = (-(0) + sqrt( (0)^2 - 4 * (-3)*(9)))/(2*-3) and x2 = (-(0) - sqrt( (0)^2 - 4 * (-3)*(9)))/(2*-3)
Which can be turned into x1 = ( 0 + sqrt( 0+108))/(-6) and x2 = ( 0 - sqrt( 0+108))/(-6)
Which make x1 = ( 0 + sqrt( 108))/(-6) and x2 = ( 0 - sqrt( 108))/(-6)
We can get x1 = ( 10.3923048454133 )/(-6) and x2 = ( - 10.3923048454133 )/(-6)
So we have the answers x1 = -1.73205080756888 and x2 = 1.73205080756888

1B. Use completing the square to find the root of the equation !
-3*x^2 + 9 = 0 ,divide both side with -3
Which result in x^2 - 3 = 0 ,
Which means that the coefficient of x is 0
We have to use the fact that ( x + q )^2 = x^2 + 2*q*x + q^2 , and assume that q = 0/2 = 0
Which means we can turn the equation into x^2 - 3 = 0
Which can be turned into ( x )^2 - 3 = 0
Which is the same with (( x ) - 1.73205080756888 ) * (( x ) + 1.73205080756888 ) = 0
And it is the same with ( x - 1.73205080756888 ) * ( x + 1.73205080756888 ) = 0
Do the addition/subtraction, and we get ( x - 1.73205080756888 ) * ( x + 1.73205080756888 ) = 0
The answers are x1 = 1.73205080756888 and x2 = -1.73205080756888

-x^2+4=2x^2-5

-3x^2=-9

x^2=3
x=+/- sq rt 3