Using the discriminant, how many real number solutions does this equation have?
3x^2 – 2 = 5x
3 x² – 2 = 5 x
3 x² – 2 - 5 x = 0
3 x² - 5 x - 2 = 0
a x² + b x + c = 0
in this case:
a = 3 , b = - 5 , c = - 2
D = b² - 4 ∙ a ∙ c
D = ( - 5 )² - 4 ∙ 3 ∙ ( - 2 )
D = 25 + 4 ∙ 3 ∙ 2
D = 25 + 24
D = 49
If D > 0, the equation has 2 real solutions.
If D = 0, the equation has 1 real solution.
If D < 0, the equation has 2 conjugate imaginary solutions.
In this case D > 0 , so the equation has 2 real solutions.
To determine the number of real number solutions of the quadratic equation 3x^2 - 5x - 2 = 0, we can use the discriminant. The discriminant is part of the quadratic formula and can be calculated using the formula:
Discriminant (D) = b^2 - 4ac
In this equation:
a = 3
b = -5
c = -2
Now, we can substitute these values into the discriminant formula:
D = (-5)^2 - 4 * 3 * (-2)
= 25 - (-24)
= 25 + 24
= 49
Now that we have found the discriminant (D = 49), we can determine the number of real number solutions:
1. If the discriminant (D) is positive (D > 0), then the quadratic equation has two distinct real number solutions.
2. If the discriminant (D) is zero (D = 0), then the quadratic equation has one real number solution (a repeated root).
3. If the discriminant (D) is negative (D < 0), then the quadratic equation has no real number solutions.
In this case, the discriminant (D) is positive (D = 49), which means the quadratic equation 3x^2 - 5x - 2 = 0 has two distinct real number solutions.