Solve the following equation using the quadratic formula: [tex]2 x^2-7 x=1[/tex]
The solution set is $\square$ . (Simplify your answer, removing any radicals and [tex]i[/tex] as needed. Use a comma to separate answers as needed)
Answer (1)
Rewrite the equation in standard form: 2 x 2 − 7 x − 1 = 0 .
Identify coefficients: a = 2 , b = − 7 , c = − 1 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Simplify to find the solutions: x = 4 7 ± 57 , thus the solution set is 4 7 − 57 , 4 7 + 57 .
Explanation
Rewrite the equation First, we need to rewrite the given equation 2 x 2 − 7 x = 1 in the standard quadratic form a x 2 + b x + c = 0 . Subtracting 1 from both sides, we get 2 x 2 − 7 x − 1 = 0 .
Identify coefficients Now, we identify the coefficients: a = 2 , b = − 7 , and c = − 1 .
State the quadratic formula We will use the quadratic formula to solve for x : x = 2 a − b ± b 2 − 4 a c
Substitute and simplify Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 2 ) − ( − 7 ) ± ( − 7 ) 2 − 4 ( 2 ) ( − 1 ) x = 4 7 ± 49 + 8 x = 4 7 ± 57
Find the solutions Therefore, the two solutions are: x 1 = 4 7 − 57 x 2 = 4 7 + 57
State the solution set The solution set is { 4 7 − 57 , 4 7 + 57 } .
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 determine its maximum height and range. Architects and engineers use quadratic equations to design arches and bridges, ensuring structural stability and optimal performance. Financial analysts also employ quadratic equations to model investment growth and predict market trends, aiding in making informed decisions.
