[tex]$3 x^2+23 x-8=0$[/tex]
Answer (2)
Identify the quadratic equation: 3 x 2 + 23 x − 8 = 0 .
Calculate the discriminant using D = b 2 − 4 a c : D = 2 3 2 − 4 ( 3 ) ( − 8 ) = 625 .
Apply the quadratic formula: x = 2 a − b ± D = 6 − 23 ± 625 .
Determine the two solutions: x = 3 1 and x = − 8 .
Explanation
Understanding the Problem and the Quadratic Formula We are given the quadratic equation 3 x 2 + 23 x − 8 = 0 . Our goal is to find the solutions (roots) of this equation. We can use the quadratic formula to solve for x . The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c , where a = 3 , b = 23 , and c = − 8 .
Calculating the Discriminant First, we need to calculate the discriminant, which is the part under the square root in the quadratic formula: D = b 2 − 4 a c . Substituting the values a = 3 , b = 23 , and c = − 8 , we get: D = ( 23 ) 2 − 4 ( 3 ) ( − 8 ) = 529 + 96 = 625
Applying the Quadratic Formula Now we substitute the values of a , b , and c into the quadratic formula: x = 2 ( 3 ) − 23 ± 625 = 6 − 23 ± 25
Finding the Solutions We have two possible solutions for x :
x 1 = 6 − 23 + 25 = 6 2 = 3 1
x 2 = 6 − 23 − 25 = 6 − 48 = − 8
So the solutions are x = 3 1 and x = − 8 .
Final Answer Therefore, the solutions to the quadratic equation 3 x 2 + 23 x − 8 = 0 are x = 3 1 and x = − 8 .
Examples
Quadratic equations are incredibly useful in various real-world scenarios, such as calculating the trajectory of a ball, designing bridges, or determining the optimal dimensions for a garden to maximize area. For example, if you're launching a rocket, you need to know its path, which can be modeled by a quadratic equation. Similarly, engineers use quadratic equations to ensure structures are stable and efficient. Understanding how to solve these equations helps in making informed decisions and precise calculations in these fields.
To solve the equation 3 x 2 + 23 x − 8 = 0 , we use the quadratic formula resulting in solutions x = 3 1 and x = − 8 . First, we calculate the discriminant as 625 , then find the roots using the formula. Thus, the two solutions to the equation are x = 3 1 and x = − 8 .
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