3x^2-24x=18

Question

Solve.if no real numbers exist state this

Answer 1

Subtract 18 on both sides of the equation.

 

3x² - 24x - 18 = 0

 

 

We can factor out a 3.

 

3(x² - 8x - 6) = 0

 

 

x² - 8x - 6 = 0

 

 

To determine whether x is a real number or not, we look at the discriminant of the quadratic formula:

 

 

Quadratic formula

 

x = (-b ± √(b² - 4ac)) / 2a

 

where:

a = 1

b = -8

c = -6

 

 

Discriminant

 

b² - 4ac

 

 

Real number:  b² ≥ 4ac

Complex number:  b² < 4ac

 

b² = (-8)² = 64

 

4ac = 4(1)(-6) = -24

 

 

This indicates that x is a real number.  You can go on ahead and solve using the quadratic formula.

Answer 2

3x² - 24x = 18

 

3x² - 24x - 18 = 0

 

Divide both sides by 3:

 

      x² - 8x - 6 = 0

 

Let's look at the discriminant, b² - 4ac, to see how many real solutions there are:

 

      b² - 4ac = (-8)² - 4(1)(-6) = 88

 

Because the discriminant is greater than zero, there are two real solutions.  Use the quadratic formula to find the solutions:

 

      x = [-b±√(b²-4ac)]/2a

 

      x = 4 ± √22