Use the discriminant to determine how many real number solutions the equation has
1-7a^2=-7a-2
oops i mean 2...sry
To determine the number of real number solutions of the equation 1-7a^2=-7a-2 using the discriminant, we need to first rewrite the equation in quadratic form.
1-7a^2 = -7a - 2
Rearranging the terms,
7a^2 - 7a - 3 = 0
Now, let's identify the coefficients of the quadratic equation:
a = 7
b = -7
c = -3
The discriminant (D) is given by the formula:
D = b^2 - 4ac
Substituting the values,
D = (-7)^2 - 4 * 7 * (-3)
D = 49 + 84
D = 133
Now, let's analyze the value of the discriminant:
If D > 0, then the quadratic equation has two distinct real solutions.
If D = 0, then the quadratic equation has one real solution (a double root).
If D < 0, then the quadratic equation has no real solutions (only complex solutions).
In this case, D = 133, therefore D > 0. This means that the quadratic equation has two distinct real solutions.
To determine the number of real number solutions for the equation 1-7a^2 = -7a-2, we can use the discriminant. The discriminant is a mathematical formula that helps us determine the nature of solutions of a quadratic equation.
The given equation can be rearranged into a quadratic equation form: 7a^2 - 7a - 3 = 0
The discriminant (denoted as Δ) of a quadratic equation ax^2 + bx + c = 0 is given by the formula: Δ = b^2 - 4ac
In our equation, a = 7, b = -7, and c = -3.
We can substitute these values into the discriminant formula:
Δ = (-7)^2 - 4(7)(-3)
= 49 + 84
= 133
Now, based on the value of the discriminant:
1. If Δ > 0, there are two distinct real solutions.
2. If Δ = 0, there is one repeated real solution.
3. If Δ < 0, there are no real solutions.
In our case, since Δ = 133, which is greater than 0, the equation has two distinct real number solutions.