Question

Decide how many solutions the equation has. $$x^{2}+16 x+64=0$$

Short answer

The quadratic equation has exactly one real solution. This is derived from the fact that the value of the discriminant equals to 0.

Step-by-step solution

Identify coefficients

In the quadratic equation \(x^2 + 16x + 64 = 0\), the coefficients of x^2, x, and the constant term are a = 1, b = 16, and c = 64 respectively. We identify these coefficients to apply in the discriminant formula.

Apply the Discriminant Formula

Apply the coefficients into the discriminant formula \(D = b^2 - 4ac\). So D equals to \( (16)^2 - 4*1*64 \). Calculate the new value of D.

Calculate the Discriminant and Determine the Number of Solutions

D equals to 256 - 256 which equals to 0. Given the value of the discriminant, if D > 0, there are two distinct solutions, if D = 0, there is one real solution, and if D < 0, there aren't any real solutions. Thus, in our case, as the discriminant equals to 0, there is exactly one real solution to this quadratic equation.