Question

show that the fundamental theorem of algebra is true for the quadratic polynomial - 4x² - 24x - 36 = 0 by using the quadratic formula. which of the following statements accurately describes the solution set? (1 point) there are two identical solutions there are two rational solutions there are two non - real solutions there are two irrational solutions

Explanation:

Step1: Identify coefficients

For the quadratic equation \(-4x^{2}-24x - 36=0\), rewrite it in standard form \(ax^{2}+bx + c = 0\), so \(a=-4\), \(b=-24\), \(c = - 36\).

Step2: Calculate the discriminant

The discriminant \(\Delta=b^{2}-4ac\). Substitute \(a=-4\), \(b=-24\), \(c=-36\) into it: \(\Delta=(-24)^{2}-4\times(-4)\times(-36)=576 - 576=0\).

Step3: Analyze the nature of solutions

When \(\Delta = 0\), the quadratic - formula \(x=\frac{-b\pm\sqrt{\Delta}}{2a}\) gives \(x=\frac{-(-24)\pm\sqrt{0}}{2\times(-4)}=\frac{24\pm0}{-8}\). There is one real - valued solution (or two identical real - valued solutions) which is rational since \(x = - 3\).

Answer:

There are two identical solutions