Question

find the discriminant.
\\( 3s = -5s^2 \\)
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 the equation \(3s=- 5s^{2}\), we can rewrite it as \(5s^{2}+3s = 0\), where \(a = 5\), \(b = 3\), and \(c = 0\).

Step2: Recall the formula for the discriminant

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

Step3: Calculate the discriminant

Substitute \(a = 5\), \(b = 3\), and \(c = 0\) into the discriminant formula:
\(D=(3)^{2}-4\times5\times0\)
\(D = 9-0\)
\(D=9\)

Step4: Determine the type of solutions

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

Since \(D = 9>0\), the equation has two real solutions.

Answer:

The discriminant is \(9\) and the equation has two real solutions.