Question

directions: in 15 - 18, find the discriminant of each equation and determine the number and type of roots. then solve using the most appropriate method. use each method once only. 15. $-x^2 + 2x - 8 = 0$
discriminant:

of roots:

type of roots:
method for solving:

Explanation:

Step1: Rewrite the equation in standard form

The given equation is \(-x^{2}+2x - 8=0\). Multiply both sides by \(- 1\) to get \(x^{2}-2x + 8=0\). For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b=-2\), \(c = 8\)), the discriminant \(D=b^{2}-4ac\).

Step2: Calculate the discriminant

Substitute \(a = 1\), \(b=-2\), \(c = 8\) into the discriminant formula:
\(D=(-2)^{2}-4\times1\times8=4 - 32=- 28\)

Step3: Determine the number and type of roots

Since the discriminant \(D=-28<0\), the number of roots is \(2\) (in the complex number system), and the type of roots is complex (non - real) roots (specifically, they are complex conjugates of the form \(a\pm bi\) where \(a=\frac{-b}{2a}\) and \(b=\frac{\sqrt{\vert D\vert}}{2a}\)).

Step4: Solve the quadratic equation

We can use the quadratic formula \(x=\frac{-b\pm\sqrt{D}}{2a}\). Since \(D=-28=\sqrt{-28}i = 2\sqrt{7}i\) (where \(i=\sqrt{-1}\)), and \(a = 1\), \(b=-2\), we have:
\(x=\frac{-(-2)\pm\sqrt{-28}}{2\times1}=\frac{2\pm2\sqrt{7}i}{2}=1\pm\sqrt{7}i\)

Answer:

• Discriminant: \(-28\)
• \(\#\) of Roots: \(2\)
• Type of Roots: Complex (non - real) roots
• Method for Solving: Quadratic formula