Match each equation with its number of unique solutions.
[tex]y=3 x^2-6 x+3 \quad y=-2 x^2+9 x-11 \quad y=-x^2-4 x+7[/tex]
| Two Real Solutions | One Real Solution |
|---|---|
| | |
| One Complex Solution | Two Complex Solutions |
| | |
Answer (2)
The first equation has one real solution, the second equation has two complex solutions, and the third equation has two real solutions. The matches are: y = 3 x 2 − 6 x + 3 - One Real Solution, y = − 2 x 2 + 9 x − 11 - Two Complex Solutions, y = − x 2 − 4 x + 7 - Two Real Solutions.
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Calculate the discriminant D = b 2 − 4 a c for each quadratic equation.
If 0"> D > 0 , there are two real solutions; if D = 0 , there is one real solution; if D < 0 , there are two complex solutions.
For y = 3 x 2 − 6 x + 3 , D = 0 , so one real solution.
For y = − 2 x 2 + 9 x − 11 , D = − 7 , so two complex solutions.
For y = − x 2 − 4 x + 7 , D = 44 , so two real solutions.
Explanation
Understanding the Problem We are given three quadratic equations and asked to determine the number of unique solutions (real or complex) for each. To do this, we will calculate the discriminant ( D = b 2 − 4 a c ) for each equation. The discriminant tells us about the nature of the roots:
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 (conjugate pairs).
Calculating Discriminant 1 For the first equation, y = 3 x 2 − 6 x + 3 , we have a = 3 , b = − 6 , and c = 3 . The discriminant is: D 1 = ( − 6 ) 2 − 4 ( 3 ) ( 3 ) = 36 − 36 = 0
Interpreting Discriminant 1 Since D 1 = 0 , the first equation has one real solution.
Calculating Discriminant 2 For the second equation, y = − 2 x 2 + 9 x − 11 , we have a = − 2 , b = 9 , and c = − 11 . The discriminant is: D 2 = ( 9 ) 2 − 4 ( − 2 ) ( − 11 ) = 81 − 88 = − 7
Interpreting Discriminant 2 Since D 2 = − 7 < 0 , the second equation has two complex solutions.
Calculating Discriminant 3 For the third equation, y = − x 2 − 4 x + 7 , we have a = − 1 , b = − 4 , and c = 7 . The discriminant is: D 3 = ( − 4 ) 2 − 4 ( − 1 ) ( 7 ) = 16 + 28 = 44
Interpreting Discriminant 3 Since 0"> D 3 = 44 > 0 , the third equation has two real solutions.
Final Answer Therefore, the equations match to the number of unique solutions as follows:
y = 3 x 2 − 6 x + 3 : One Real Solution
y = − 2 x 2 + 9 x − 11 : Two Complex Solutions
y = − x 2 − 4 x + 7 : Two Real Solutions
Examples
Understanding the discriminant of a quadratic equation is crucial in many real-world applications. For example, when designing a bridge, engineers use quadratic equations to model the arch's shape. Knowing whether the equation has real roots helps determine if the arch will intersect the ground at stable points. Similarly, in physics, analyzing projectile motion involves quadratic equations, and the discriminant can tell us if a projectile will hit a target (real roots) or not (complex roots). These concepts are also used in optimization problems in economics and computer science to find maximum or minimum values.
