Solve by the quadratic formula.
[tex]12 x^2-10 x-8=0[/tex]
Answer (1)
Identify the coefficients: a = 12 , b = − 10 , c = − 8 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 24 10 ± 484 = 24 10 ± 22 .
Solve for x : x 1 = 3 4 and x 2 = − 2 1 . The final answer is − 2 1 , 3 4 .
Explanation
Understanding the Problem We are given the quadratic equation 12 x 2 − 10 x − 8 = 0 and asked to solve for x using the quadratic formula. The quadratic formula is a general method for finding the roots of any quadratic equation of the form a x 2 + b x + c = 0 .
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 .
Identifying Coefficients In our equation, 12 x 2 − 10 x − 8 = 0 , we can identify the coefficients as follows: a = 12 b = − 10 c = − 8
Substituting Values Now, we substitute these values into the quadratic formula: x = 2 ( 12 ) − ( − 10 ) ± ( − 10 ) 2 − 4 ( 12 ) ( − 8 )
Simplifying Simplify the expression: x = 24 10 ± 100 + 384 x = 24 10 ± 484 x = 24 10 ± 22
Calculating the Roots Now we find the two possible values for x :
x 1 = 24 10 + 22 = 24 32 = 3 4 x 2 = 24 10 − 22 = 24 − 12 = − 2 1
Final Answer Therefore, the solutions for the quadratic equation 12 x 2 − 10 x − 8 = 0 are x = 3 4 and x = − 2 1 .
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. Engineers use quadratic equations to design bridges and arches, ensuring structural stability. Financial analysts also employ them to predict profits and losses, optimizing investment strategies. Understanding quadratic equations provides valuable tools for problem-solving in physics, engineering, and finance.
