solve the equation 5x(x-5)=(4x+9)(x-5)

Question

solve the equation 5x(x-5)=(4x+9)(x-5)

Answer 1

Their are 2 answers X = 5, and X = 9

 

 x = 5 because on both sides of your equation you have (x-5) in parenthesis and you are multiplying that number by the rest of the equation.  If X = 5, then your last step on both sides of the equation will be to multiply by 0.  Hence x=5.

 

Now X = 9.  This solution will take a bit more work to solve first you will need to distribute both sides of your equation

 

5x(x-5) - to distribute this side of the equation simply multiply both numbers in the parenthesis by 5x (remember the subtraction sign means you will have to multiply by a negative number. So it becomes

 

(5x * x) + (5x * -5) Or 5x² + -25x

 

The other side (4x + 9)(x -5) needs to be distributed.  I tend to use the FOIL method, but the box method works well too.

 

(4x * x) + (4x * -5) + (9 * x) + (9 x -5)

    4x²    +   -20x    +     9x    +   -45     Next add together like terms, remember you can't add an x to an x²

    4x^{2                     + -11x          +    -45}

 

Okay now that we have distributed both sides I'm going to put both sides of our equation down distributed out.

 

5x² + -25x = 4x² -11x + -45

 

To solve for x, I need all the x's on one side of the equation, so I will use the inverse operations (remember the inverse of addition is subtraction and the inverse of multiplication is division).  Whatever I do to one side of the equation has to be done to the other side to keep it balanced.

 

  5x² + -25x = 4x² -11x + -45

 -4x²            -4x²

   x^{2    + -25x       0   -11x + -45}

          +11x         +11x

    x² + -14x  + 0 + 0 + -45

 

      x² + -14x = - 45

 

Now that we have x on one side, the next step is a little bit tricky.

 

First we need to divide the number next to x by 2, So -14 ÷ 2 = -7

Now we need square -7.  -7² = 49.  We are doing all of this because we want one side of our equation to equal (x-7)²

 

Now we need to add 49 to both sides of our equation.

 

x² + -14x + 49 = -45 + 49

 

x² + -14x + 49 = 4

 

 

Now remember we had to do some work to find 49 by dividing and squaring a number.  We did all of that because we needed to find the distributed version of (x - 7)²  If your nervous about this step you can always foil it out to check your work

 

(x-7)(x-7)

 

(x*x) + (x * -7) + (-7 * x) + (-7 * -7)

   X²  +      -7x   +    - 7x  +      49

   X²  + -14x + 49   My answer checks.  I always double check this step because if I did any of the math wrong, then my final answer will be wrong.

 

Now my equation reads

 

(x - 7)² = 4

 

I need to take the square root of both sides of this equation

 

√(x-7)² = √4

 

   x-7 = 2

 

Now to finish solving for X, simply get X by itself using the inverse operation.

 

x - 7 = 2

  + 7  +7

 x - 0 = 9

 

   x = 9

 

Be sure to check your work by replacing x in your original equation.

 

5*9(9-5)= (4*9+9)(9-5)

5*9(4)    = (36 +9)(4)

  45 (4)         (45) (4)

   192      =       192

 

If you made any mistakes in your problem you can go back and find them if your final check does not work.  Best of luck

 

- Dannielle Cavanaugh - Rouse

 

 

 

 

Answer 2

x=5 or 9.

E.H.