Question

1. use the discriminant to describe the number of solutions of the equation $2x^2 + 5x - 12 = 0$

• the discriminant is 121, and the equation has two real, rational roots.
• the discriminant is 121, and the equation has one real, rational root.
• the discriminant is -71, and the equation has two real, irrational roots.
• the discriminant is -71, and the equation has two complex roots.

Explanation:

Step1: Recall the discriminant formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the discriminant \(D\) is given by \(D=b^{2}-4ac\). In the equation \(2x^{2}+5x - 12 = 0\), we have \(a = 2\), \(b = 5\), and \(c=- 12\).

Step2: Calculate the discriminant

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:

$$
LATEXBLOCK0
$$

Step3: Analyze the nature of the roots

Since \(D = 121>0\) and \(121\) is a perfect square (\(\sqrt{121}=11\)), the quadratic equation has two real, rational roots.

Answer:

The discriminant is 121, and the equation has two real, rational roots.