Determine the value of the discriminant and indicate the number of real solutions for the quadratic.
[tex]x^2-3 x+7=0[/tex]
Discriminant: $\square$
Are the solutions real or complex? $\square$
How many solutions are there? $\square
Answer (2)
The discriminant of the quadratic equation x 2 − 3 x + 7 = 0 is − 19 , indicating that there are no real solutions. Instead, the solutions are complex. Therefore, the number of real solutions is 0 .
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Identify the coefficients of the quadratic equation: a = 1 , b = − 3 , and c = 7 .
Calculate the discriminant using the formula D = b 2 − 4 a c : D = ( − 3 ) 2 − 4 ( 1 ) ( 7 ) = − 19 .
Determine the nature of the solutions based on the discriminant: Since D < 0 , the solutions are complex.
State the number of real solutions: There are 0 real solutions, so the final answer is 0 .
Explanation
Problem Analysis We are given the quadratic equation x 2 − 3 x + 7 = 0 . Our goal is to find the discriminant, determine the nature of the solutions (real or complex), and find the number of real solutions.
Identifying Coefficients The general form of a quadratic equation is a x 2 + b x + c = 0 . In our equation, we can identify the coefficients as follows:
a = 1 b = − 3 c = 7
Discriminant Formula The discriminant, denoted by D , is given by the formula:
D = b 2 − 4 a c
Calculating the Discriminant Now, we substitute the values of a , b , and c into the discriminant formula:
D = ( − 3 ) 2 − 4 ( 1 ) ( 7 ) D = 9 − 28 D = − 19
Determining the Nature of Solutions The nature of the solutions is determined by the value of the discriminant:
If 0"> 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 solutions.
Since D = − 19 < 0 , the equation has two complex solutions. Therefore, the number of real solutions is 0.
Final Answer The discriminant is -19, the solutions are complex, and there are 0 real solutions.
Discriminant: − 19
Are the solutions real or complex? C o m pl e x
How many solutions are there? 0
Examples
Understanding the discriminant helps us predict the type of solutions we'll get when solving quadratic equations. For example, if we're designing a bridge and modeling its structure with a quadratic equation, a negative discriminant would tell us that our model predicts no real-world solutions for certain parameters, indicating a potential flaw in the design or the model itself. This knowledge allows engineers to adjust designs or models before physical construction, saving time and resources. Similarly, in physics, when analyzing projectile motion, the discriminant can reveal whether a projectile will hit a target at a certain height, informing necessary adjustments to launch angles or initial velocities.
