Question

find the discriminant.
\\(9r^2 = -2r\\)
what type of solutions does the equation have?
one real solution
two real solutions
two complex (non-real) solutions

Explanation:

Step1: Rewrite the equation in standard form

The standard form of a quadratic equation is \(ax^{2}+bx + c = 0\). Given \(9r^{2}=-2r\), we add \(2r\) to both sides to get \(9r^{2}+2r = 0\). Here, \(a = 9\), \(b = 2\), and \(c = 0\).

Step2: Recall the discriminant formula

The discriminant \(D\) of a quadratic equation \(ax^{2}+bx + c = 0\) is given by \(D=b^{2}-4ac\).

Step3: Calculate the discriminant

Substitute \(a = 9\), \(b = 2\), and \(c = 0\) into the discriminant formula:
\(D=(2)^{2}-4\times9\times0\)
\(= 4 - 0\)
\(= 4\)

Step4: Determine the type of solutions

• If \(D>0\), the equation has two distinct real solutions.
• If \(D = 0\), the equation has one real solution (a repeated root).
• If \(D<0\), the equation has two complex (non - real) solutions.

Since \(D = 4>0\), the equation has two real solutions.

Answer:

The discriminant is \(4\), and the equation has two real solutions.