Select the correct answer from each drop-down menu. Consider the quadratic equation $2 x^2+3 x+5=0$. The discriminant of this quadratic equation is $\square$ zero. This means the quadratic equation will have $\square$ real solution(s) and $\square$ complex solution(s).
Answer (2)
The discriminant of the quadratic equation 2 x 2 + 3 x + 5 = 0 is − 31 , which is less than zero. This means the quadratic equation will have 0 real solutions and 2 complex solutions.
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Calculate the discriminant D using the formula D = b 2 − 4 a c , where a = 2 , b = 3 , and c = 5 , resulting in D = − 31 .
Since D < 0 , the quadratic equation has no real solutions.
Because the equation has no real solutions, it has two complex solutions.
The quadratic equation 2 x 2 + 3 x + 5 = 0 has a discriminant of − 31 , 0 real solutions, and 2 complex solutions, so the final answer is 0 real solution(s) and 2 complex solution(s) .
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 + 3 x + 5 = 0 . Our goal is to find the discriminant of this equation and determine how many real and complex solutions it has.
Calculating the Discriminant The discriminant of a quadratic equation in the form a x 2 + b x + c = 0 is given by the formula D = b 2 − 4 a c . In our equation, a = 2 , b = 3 , and c = 5 . Let's calculate the discriminant: D = b 2 − 4 a c = ( 3 ) 2 − 4 ( 2 ) ( 5 ) = 9 − 40 = − 31
Interpreting the Discriminant The discriminant is − 31 . Now, we need to determine the number of real and complex solutions based on the value of the discriminant.
Rules for Solutions
If 0"> 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 no real solutions, but it has two complex solutions.
Determining the Number of Solutions Since our discriminant D = − 31 is less than zero, the quadratic equation has no real solutions and two complex solutions.
Final Answer Therefore, the discriminant of the quadratic equation 2 x 2 + 3 x + 5 = 0 is − 31 , which is less than zero. This means the quadratic equation will have 0 real solutions and 2 complex solutions.
Examples
Understanding the discriminant helps us predict the nature of solutions in various real-world scenarios. For example, in engineering, when designing a bridge, the quadratic equations that model the structure's stability can be analyzed using the discriminant. A negative discriminant would indicate that the structure, as modeled, might not have real-world stability, prompting a redesign. Similarly, in physics, analyzing projectile motion involves quadratic equations, and the discriminant can tell us whether a projectile will actually hit a target or if the conditions (like initial velocity and angle) need adjustment. This ensures designs are viable and safe before implementation.
