How to find real solutions of an equation?

Question

1.) x^2-6x=27

2.) x^3+3x^2-x-3=0

3.) x^3+x^2-16x=16

4.) 2x^3-3x^2-18x+27=0

Answer 1

1)

 

x² - 6x = 27

 

 

Subtract 27 on both sides of equation.

 

x² - 6x - 27 = 0

 

 

Factor the left side of equation using FOIL.

 

(x + 3)(x - 9) = 0

 

 

Set the factors equal to zero.

 

x + 3 = 0            and           x - 9 = 0

 

x = -3                 and           x = 9

 

 

We have two solutions.

 

x = -3 

x = 9

 

 

2)

 

x³ + 3x² - x - 3 = 0

 

 

We use the possible roots of the polynomial to factor out the left side.  The possible roots are -3, -1, 1, and 3.  Lets make guess and choose -3 as our possible root.  Then divide the left side of equation by the factor that gives that root using long division or synthetic division.  If the quotient gets us a remainder of zero, then that quotient is also a factor.  If we follow the process, we will see that -3 is a root of the left side. 

 

(x + 3)(x² - 1) = 0

(x + 3)(x + 1)(x - 1) = 0

 

 

Set the factors equal to zero.

 

x + 3 = 0           x + 1 = 0             x - 1 = 0

 

x = -3                x = -1                  x = 1

 

 

We have three solutions.

 

x = -3

x = -1

x = 1

 

 

3)

 

x³ + x² - 16x = 16

 

 

Subtract 16 on both sides of equation. 

 

x³ + x² - 16x - 16 = 0

 

 

Factor the left side by grouping.

 

x²(x + 1) - 16(x + 1) = 0

 

 

The GCF of the left side is  (x + 1).

 

(x + 1)(x² - 16) = 0

(x + 1)(x + 4)(x - 4) = 0

 

 

x + 1 = 0               x + 4 = 0                    x - 4 = 0

x = -1                    x = -4                         x = 4

 

 

We have three solutions.

 

x = -1

x = -4

x = 4

 

 

3)

 

2x³ - 3x² - 18x + 27 = 0

 

 

Factor by grouping, just like in the last problem.

 

x²(2x - 3) - 3(6x - 9) = 0

 

x²(2x - 3) - 3*3(2x - 3) = 0

 

x²(2x - 3) - 9(2x - 3) = 0

 

 

The GCF is  (2x - 3).

 

(2x - 3)(x² - 9) = 0

 

(2x - 3)(x + 3)(x - 3) = 0

 

 

If we set the factors equal to zero, we will obtain the three solutions.

 

x = -3

x = 3/2

x = 3