Solve the quadratic equation $3 x^2+x-5=0$. Give your answers to 2 decimal places.
Answer (1)
Identify the coefficients: a = 3 , b = 1 , c = − 5 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 6 − 1 ± 61 .
Calculate and round the solutions: x 1 ≈ 1.14 , x 2 ≈ − 1.47 .
x 1 ≈ 1.14 , x 2 ≈ − 1.47
Explanation
Understanding the Problem We are given the quadratic equation 3 x 2 + x − 5 = 0 . Our goal is to find the solutions for x , rounded to two decimal places. We will use the quadratic formula to solve for x .
Identifying Coefficients and the 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 = 3 , b = 1 , and c = − 5 .
Substituting Values into the Formula Now, we substitute the values of a , b , and c into the quadratic formula: x = 2 ( 3 ) − 1 ± 1 2 − 4 ( 3 ) ( − 5 )
Simplifying the Expression Next, we simplify the expression under the square root: 1 2 − 4 ( 3 ) ( − 5 ) = 1 + 60 = 61
Calculating the Square Root So, we have: x = 6 − 1 ± 61
Calculating the Roots We know that 61 ≈ 7.8102 . Therefore, we can calculate the two possible values for x :
x 1 = 6 − 1 + 7.8102 ≈ 6 6.8102 ≈ 1.1350 x 2 = 6 − 1 − 7.8102 ≈ 6 − 8.8102 ≈ − 1.4684
Rounding to Two Decimal Places Finally, we round the solutions to two decimal places: x 1 ≈ 1.14 x 2 ≈ − 1.47
Final Answer The solutions to the quadratic equation 3 x 2 + x − 5 = 0 , rounded to two decimal places, are x 1 ≈ 1.14 and x 2 ≈ − 1.47 .
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 instance, if you're launching a rocket, you can use a quadratic equation to predict its path, ensuring it reaches its intended target safely and accurately. Understanding how to solve these equations is crucial for making informed decisions and solving practical problems in fields like engineering, physics, and economics.
