Solve the quadratic equation [tex]3 x^2-5 x-7=0[/tex]. Give your answers to 3 significant figures.
Answer (1)
Identify the coefficients: a = 3 , b = − 5 , c = − 7 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 6 5 ± 109 .
Calculate the two solutions to 3 significant figures: x ≈ 2.57 , − 0.907 .
Explanation
Understanding the Problem We are given the quadratic equation 3 x 2 − 5 x − 7 = 0 . We need to find the solutions for x , rounded to 3 significant figures. The quadratic formula is x = 2 a − b ± b 2 − 4 a c for an equation of the form a x 2 + b x + c = 0 .
Identifying Coefficients Identify the coefficients a , b , and c in the quadratic equation 3 x 2 − 5 x − 7 = 0 . In this case, a = 3 , b = − 5 , and c = − 7 .
Applying the Quadratic Formula Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c . Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 3 ) − ( − 5 ) ± ( − 5 ) 2 − 4 ( 3 ) ( − 7 ) .
Simplifying the Expression Simplify the expression: x = 6 5 ± 25 + 84 . Further simplification: x = 6 5 ± 109 .
Calculating the Values of x Calculate the two possible values for x : x 1 = 6 5 + 109 and x 2 = 6 5 − 109 . Approximating the values, we get x 1 ≈ 6 5 + 10.440 ≈ 2.573 and x 2 ≈ 6 5 − 10.440 ≈ − 0.907 .
Final Answer Therefore, the two solutions for x , rounded to three significant figures, are x 1 ≈ 2.57 and x 2 ≈ − 0.907 .
Examples
Quadratic equations are incredibly useful in various fields, such as physics, engineering, and economics. For instance, when designing a bridge, engineers use quadratic equations to calculate the curve of suspension cables. Similarly, in economics, quadratic equations can model cost and revenue to determine profit maximization points. Understanding how to solve these equations allows professionals to make informed decisions and optimize outcomes in their respective fields.
