Use the discriminant to determine the number of real solutions of 4x^2+3x-7=0 which of the following statements gives the correct explanation? (1 point)
There will be no real solutions since the discriminant is negative.
There will be two real solutions since the discriminant is positive.
There will be no real solutions since the discriminant is zero.
There will be one real solution since the discriminant is zero.
The correct statement is: There will be two real solutions since the discriminant is positive.
To determine the number of real solutions of a quadratic equation in the form ax^2 + bx + c = 0 using the discriminant, we use the following rule:
- If the discriminant (denoted as Δ) is greater than zero (Δ > 0), there are two distinct real solutions.
- If the discriminant is equal to zero (Δ = 0), there is exactly one real solution.
- If the discriminant is less than zero (Δ < 0), there are no real solutions.
In the given equation 4x^2 + 3x - 7 = 0, the discriminant can be calculated using the formula Δ = b^2 - 4ac. Therefore:
a = 4
b = 3
c = -7
Plugging these values into the discriminant formula:
Δ = (3)^2 - 4(4)(-7)
Δ = 9 + 112
Δ = 121
Since the discriminant is positive (Δ > 0), the correct statement would be:
"There will be two real solutions since the discriminant is positive."
To determine the number of real solutions for the quadratic equation 4x^2 + 3x - 7 = 0, we can use the discriminant, which is the expression inside the square root of the quadratic formula. The discriminant is given as Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
In this case, the equation is 4x^2 + 3x - 7 = 0, so we can identify a = 4, b = 3, and c = -7. Calculating the discriminant, we have:
Δ = (3)^2 - 4(4)(-7)
= 9 + 112
= 121
The value of the discriminant is positive (Δ > 0). According to the properties of the discriminant:
1. If Δ > 0, there are two distinct real solutions for the quadratic equation.
2. If Δ = 0, there is one real solution (a repeated/rooted solution) for the quadratic equation.
3. If Δ < 0, there are no real solutions (only complex solutions) for the quadratic equation.
Therefore, the correct statement is: "There will be two real solutions since the discriminant is positive."