Question

algebra 2a semester online practice
complete this assessment to review what youve learned. it will not count toward your grade.

use the discriminant to determine the number of real solutions of $-3 + 8x^2 + 10x = 0$. which of the following statements gives the correct explanation? (1 point)

\\(\circ\\) there will be no real solutions since the discriminant is negative.

\\(\circ\\) there will be one real solution since the discriminant is negative.

\\(\circ\\) there will be one real solution since the discriminant is zero.

\\(\circ\\) there will be two real solutions since the discriminant is positive.

remaining attempts : 3

Explanation:

Step1: Rewrite the quadratic equation

First, rewrite the given equation \(-3 + 8x^2 + 10x = 0\) in standard form \(ax^2 + bx + c = 0\). So we get \(8x^2 + 10x - 3 = 0\), where \(a = 8\), \(b = 10\), and \(c = -3\).

Step2: Calculate the discriminant

The formula for the discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is \(\Delta = b^2 - 4ac\). Substitute \(a = 8\), \(b = 10\), and \(c = -3\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Analyze the discriminant

Since the discriminant \(\Delta = 196>0\), for a quadratic equation, if the discriminant is positive, there are two distinct real solutions. Now let's check the options:

• The first option says discriminant is negative, but we calculated it as 196 (positive), so this is wrong.
• The second option says discriminant is negative (wrong, as we found it positive) and one real solution (wrong, positive discriminant means two real solutions), so this is wrong.
• The third option says discriminant is zero (wrong, our discriminant is 196) and one real solution, so this is wrong.
• The fourth option says there will be two real solutions since the discriminant is positive, which matches our calculation.

Answer:

There will be two real solutions since the discriminant is positive. (The option corresponding to this statement, assuming it's the last option in the list: "There will be two real solutions since the discriminant is positive.")