F(x)=-(x+1)^2+4 determine the nature of the roots
suppose you had f(x) = 4-x^2
The roots would be x = ±2
4-(x+1)^2 is the same graph, just shifted left by 1, so its roots are
x = -1±2 = -3,1
See
http://www.wolframalpha.com/input/?i=-%28x%2B1%29^2%2B4
Please help me to solve one/two of them, especially on how to determine the nature of roots of equation by not solving. E.g 3x sqrt-6x+9
To determine the nature of the roots of the quadratic equation f(x) = -(x + 1)^2 + 4, we can analyze the discriminant of the equation.
The discriminant, denoted by the symbol Δ, helps us determine whether the roots of a quadratic equation are real or complex and if they are real, it helps identify whether they are distinct or repeated.
The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.
In our equation f(x) = -(x + 1)^2 + 4, we have a = -1, b = 0, and c = 4.
The discriminant is given by the formula Δ = b^2 - 4ac.
Substituting the values in our equation: Δ = 0^2 - 4(-1)(4) = 0 - (-16) = 16.
Now, based on the value of the discriminant:
1. If Δ > 0, the equation has two distinct real roots.
2. If Δ = 0, the equation has one real root (also called a repeated root or a perfect square trinomial).
3. If Δ < 0, the equation has two complex roots.
Since we found Δ = 16, which is greater than 0, the equation has two distinct real roots.
Therefore, the nature of the roots of the equation f(x) = -(x + 1)^2 + 4 is two distinct real roots.