Solve the equation using the quadratic formula.
[tex]2 x^2-3 x-1=0[/tex]
The solution set is $\square$. (Simplify your answer.)
Answer (2)
Identify the coefficients: a = 2 , b = − 3 , c = − 1 .
Calculate the discriminant: b 2 − 4 a c = ( − 3 ) 2 − 4 ( 2 ) ( − 1 ) = 17 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 4 3 ± 17 .
The solution set is { 4 3 + 17 , 4 3 − 17 } .
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 − 3 x − 1 = 0 . We need to solve this equation using the quadratic formula.
Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c , where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 . In our case, a = 2 , b = − 3 , and c = − 1 .
Calculating the Discriminant First, we calculate the discriminant, which is the part under the square root in the quadratic formula: b 2 − 4 a c . Substituting the values, we get: b 2 − 4 a c = ( − 3 ) 2 − 4 ( 2 ) ( − 1 ) = 9 + 8 = 17
Applying the Quadratic Formula Now, we substitute the values of a , b , and the discriminant into the quadratic formula: x = 2 ( 2 ) − ( − 3 ) ± 17 = 4 3 ± 17
Finding the Solutions So, the two solutions are: x 1 = 4 3 + 17 x 2 = 4 3 − 17
Final Answer Therefore, the solution set is { 4 3 + 17 , 4 3 − 17 } .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 100 square meters and you know that one side is 5 meters longer than the other, you can set up a quadratic equation to find the exact dimensions of the garden. Understanding how to solve quadratic equations allows you to solve practical problems involving optimization and design.
To solve the equation 2 x 2 − 3 x − 1 = 0 using the quadratic formula, we identify the coefficients, calculate the discriminant, and apply the formula to find the roots. The solutions are 4 3 + 17 and 4 3 − 17 . Thus, the solution set is { 4 3 + 17 , 4 3 − 17 } .
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