How many real number solutions does the equation have?
0=5x^2+2x-12
This equation has two real number solutions.
Why?
We can solve for x by factoring or using the quadratic formula:
Factoring:
0=5x^2+2x-12
0=(5x-6)(x+2)
So the solutions are x = 6/5 and x = -2.
Using the quadratic formula:
0=5x^2+2x-12
x = (-b ± sqrt(b^2 - 4ac)) / 2a
x = (-2 ± sqrt(2^2 - 4(5)(-12))) / 2(5)
x = (-2 ± sqrt(244)) / 10
x = (-2 ± 2*sqrt(61)) / 10
Simplifying, we get x = 6/5 and x = -2, which are the same solutions as before.
Therefore, the equation has two real number solutions.
To determine the number of real number solutions for the equation 0 = 5x^2 + 2x - 12, we can use the discriminant. The discriminant is a mathematical expression that appears under the square root symbol (√) in the quadratic formula. It can help us determine the nature of the solutions (real, imaginary, or equal).
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In the equation, a = 5, b = 2, and c = -12. We can substitute these values into the quadratic formula and calculate the discriminant:
Discriminant (D) = b^2 - 4ac
D = (2)^2 - 4(5)(-12)
D = 4 + 240
D = 244
The discriminant in this case is positive (D > 0), meaning that the quadratic equation has two distinct real number solutions.