How many real-number solutions does the equation have?
-7x^2+6x+3=0
a) 1
b) 2***
c) none
d) infintely
correct, since the discriminant b^2-4ac is positive
To determine the number of real-number solutions for the equation -7x^2 + 6x + 3 = 0, we can use the discriminant formula. The discriminant (represented by the symbol Δ) is calculated as Δ = b^2 - 4ac, where a, b, and c represent the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
In this case, a = -7, b = 6, and c = 3. Substituting these values into the discriminant formula, we get:
Δ = (6)^2 - 4(-7)(3)
Δ = 36 + 84
Δ = 120
Now, we can determine the number of solutions based on the value of the discriminant:
- If Δ > 0, then there are two distinct real solutions.
- If Δ = 0, then there is one real solution (the equation has a repeated root).
- If Δ < 0, then there are no real solutions.
Since Δ = 120, which is greater than 0, we can conclude that the equation -7x^2 + 6x + 3 = 0 has two distinct real solutions. Therefore, the correct answer is b) 2.