Solve the quadratic equation using any method. Give exact answers - no decimals.
[tex]3 t^2-t-1=0[/tex]
Answer (1)
Identify the coefficients: a = 3 , b = − 1 , c = − 1 .
Apply the quadratic formula: t = 2 a − b ± b 2 − 4 a c .
Substitute the values: t = 2 ( 3 ) 1 ± ( − 1 ) 2 − 4 ( 3 ) ( − 1 ) .
Simplify to find the solutions: t = 6 1 ± 13 . The solutions are t = 6 1 ± 13 .
Explanation
Understanding the Problem We are given the quadratic equation 3 t 2 − t − 1 = 0 . Our goal is to find the exact solutions for t . Since the problem asks for exact answers, we will use the quadratic formula.
Identifying Coefficients and the Quadratic Formula The quadratic formula is given by t = 2 a − b ± b 2 − 4 a c , where a , b , and c are the coefficients of the quadratic equation a t 2 + b t + c = 0 . In our case, a = 3 , b = − 1 , and c = − 1 .
Substituting Values into the Formula Now, we substitute the values of a , b , and c into the quadratic formula: t = 2 ( 3 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 3 ) ( − 1 )
Simplifying the Expression Next, we simplify the expression: t = 6 1 ± 1 + 12 = 6 1 ± 13
Final Solutions Therefore, the two solutions for t are: t = 6 1 + 13 and t = 6 1 − 13
Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, they can model the trajectory of a ball thrown in the air, helping to predict its landing point. They are also used in engineering to design bridges and arches, ensuring structural stability. In finance, quadratic equations can help model investment growth and calculate optimal strategies. Understanding how to solve them provides a foundation for tackling many practical problems.
