Question

How many solutions are there to this equation 3x(x-4)+5-x=2x-7

Answer 1

Answer:

  2 solutions

Step-by-step explanation:

I like to use a graphing calculator to find solutions for equations like these. The two solutions are ...

• x = 1
• x = 4

__

To solve this algebraically, it is convenient to subtract 2x-7 from both sides of the equation:

  3x(x -4) +5 -x -(2x -7) = 0

  3x^2 -12x +5 -x -2x +7 = 0 . . . . . eliminate parentheses

  3x^2 -15x +12 = 0 . . . . . . . . . . . . collect terms

  3(x -1)(x -4) = 0 . . . . . . . . . . . . . . . factor

The values of x that make these factors zero are x=1 and x=4. These are the solutions to the equation. There are two solutions.

__

Alternate method

Once you get to the quadratic form, you can find the number of solutions without actually finding the solutions. The discriminant is ...

  d = b^2 -4ac . . . . where a, b, c are the coefficients in the form ax^2+bx+c

  d = (-15)^2 -4(3)(12) = 225 -144 = 81

This positive value means the equation has 2 real solutions.