Find the value(s) of [tex]$k$[/tex] for which the quadratic function: [tex]$f(x)=2 x^2+3 x+3$[/tex] intersects the line: [tex]$y=2 x+k$[/tex] at more than one point.
Answer (2)
Set the quadratic function equal to the line: 2 x 2 + 3 x + 3 = 2 x + k .
Rearrange the equation to form a quadratic equation in x : 2 x 2 + x + ( 3 − k ) = 0 .
Apply the discriminant condition for more than one solution: 0"> 1 2 − 4 ( 2 ) ( 3 − k ) > 0 .
Solve for k : \frac{23}{8}"> k > 8 23 .
Explanation
Problem Analysis We are given a quadratic function f ( x ) = 2 x 2 + 3 x + 3 and a line y = 2 x + k . We want to find the values of k for which the quadratic function intersects the line at more than one point. This means we need to find the values of k for which the equation 2 x 2 + 3 x + 3 = 2 x + k has more than one real solution.
Setting up the Equation To find the intersection points, we set the quadratic function equal to the line: 2 x 2 + 3 x + 3 = 2 x + k
Rearranging the Equation Rearrange the equation to form a quadratic equation in x :
2 x 2 + 3 x + 3 − 2 x − k = 0 2 x 2 + x + ( 3 − k ) = 0
Applying the Discriminant Condition For the quadratic equation to have more than one solution, the discriminant must be greater than zero. The discriminant is given by b 2 − 4 a c , where a = 2 , b = 1 , and c = 3 − k . So we have: 0"> b 2 − 4 a c > 0 0"> 1 2 − 4 ( 2 ) ( 3 − k ) > 0
Simplifying the Inequality Simplify the inequality: 0"> 1 − 8 ( 3 − k ) > 0 0"> 1 − 24 + 8 k > 0 0"> 8 k − 23 > 0
Solving for k Solve for k :
23"> 8 k > 23 \frac{23}{8}"> k > 8 23 2.875"> k > 2.875
Final Answer Therefore, the values of k for which the quadratic function intersects the line at more than one point are \frac{23}{8}"> k > 8 23 .
Examples
Imagine you are designing a curved slide in a water park. The slide's path can be modeled by a quadratic function, and you want to ensure that a straight safety bar intersects the slide at two points to provide maximum support. By adjusting the height of the safety bar (represented by 'k'), you can control the number of intersection points. If \frac{23}{8}"> k > 8 23 , the safety bar intersects the slide at two points, ensuring better support and safety for the riders. This mathematical approach helps engineers design safer and more enjoyable amusement park rides.
The quadratic function f ( x ) = 2 x 2 + 3 x + 3 intersects the line y = 2 x + k at more than one point if \frac{23}{8}"> k > 8 23 or approximately 2.875"> k > 2.875 . Therefore, values of k must be greater than 2.875 for two intersection points to occur. This is determined by examining the quadratic's discriminant.
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