Question

Which of the following is a solution of 3x2 = 7x - 3? a. negative 7 plus or minus the square root of 13 divided by 6 b. 7 plus or minus the square root of 85 divided by 6 c. negative 7 plus or minus the square root of 85 divided by 6 d. x - 7 plus or minus the square root of 13 divided by 6

Answer 1

The solutions for the equation
`3x^(2) =7x-3` are:
`x_(1)=(7+√(13))/(6) , x_(2)= (7-√(13) )/(6)`, so the option d is correct.

Explanation:

You have two options to find the solutions of the equation
`3x^(2) =7x-3`

The first is to solve it with the quadratic formula, for this, you need to follow these steps:

1. Add 3 and substract 7x from both sides
   `3x^(2) +3-7x=7x+3-3-7x`
2. Simplify
   `3x^(2) +3-7x=0`
3. Use the quadratic equation formula. For a quadratic equation of the form
   `ax^(2) +bx+c=0` the solutions are
   `x_(1,2) = \frac{-b\±\sqrt{b^(2)-4ac } }{2a}`
4. In the equation given
   `a=3, b=-7, c=3`, so
   `x_(1) = \frac{-(-7)+\sqrt{(-7)^(2)-4*3*3 } }{2*3}, x_(2) = \frac{-(-7)-\sqrt{(-7)^(2)-4*3*3 } }{2*3}`
5. The solutions are
   `x_(1)=(7+√(13))/(6) , x_(2)= (7-√(13) )/(6)`

The second option is solve the equation by completing the square:

1. Substract 7x from both sides
   `3x^(2) -7x=7x-7x-3` and simplify
   `3x^(2) -7x=-3`
2. Divide both sides by 3
   `(3x^(2)-7x )/(3)= -(3)/(3)` and simplify
   `x^(2) -(7x)/(3)=-1`
3. Use the fact that
   `x^(2) +2ax+a^(2)=(x+a)^(2)`
4. We need to find
   `a`, for this, we use this relation
   `2ax=-(7x)/(3)` and solve for
   `a`, we get
   `a = -(7)/(6)`
5. Add
   `a^(2) = ((-7)/(6) )^(2) =(13)/(36)` to both sides in the equation of step 2
   `x^(2) -(7x)/(3)+(-(7)/(6) )^(2)=-1 +(-(7)/(6) )^(2)` and simplify
   `x^(2) -(7x)/(3)+(-(7)/(6) )^(2)=(13)/(36)`
6. Complete the square, use step 3,
   `x^(2) -(7x)/(3)+(-(7)/(6) )^(2) = (x-(7)/(6) )^(2)=(13)/(36)`
7. For
   `f^(2)(x)` the solutions are
   `f(x) =√(a),-√(a)`, so
   `x-(7)/(6) = \sqrt{(13)/(36) } =(√(13)+7 )/(6)\ and \ x-(7)/(6) = -\sqrt{(13)/(36) } =-(√(13)+7 )/(6)`

Answer 2

3x^2 = 7x - 3
Rearranging,
3x^2 - 7x + 3 = 0
Using quadratic formula :
x= [7 + sqrt (49 - 36)]/6
x= [7 + sqrt 13]
or
x= [7 - sqrt (49 - 36)]/6
x= [7 - sqrt 13]/6

d. x - 7 plus or minus the square root of 13 divided by 6