Solve the equation [tex]$3 x^2+13 x-7=0$[/tex] given the answer to two decimal places.
Answer (1)
∙ Identify the coefficients a , b , and c from the quadratic equation 3 x 2 + 13 x − 7 = 0 .
∙ Calculate the discriminant D using the formula D = b 2 − 4 a c .
∙ Apply the quadratic formula x = 2 a − b ± D to find the two solutions for x .
∙ Round the solutions to two decimal places: x 1 ≈ 0.48 and x 2 ≈ − 4.82 . The solutions are 0.48 , − 4.82 .
Explanation
Problem Analysis We are given the quadratic equation 3 x 2 + 13 x − 7 = 0 . Our goal is to find the solutions for x , rounded to two decimal places. We will use the quadratic formula to solve this equation.
Quadratic Formula The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c where a = 3 , b = 13 , and c = − 7 in our equation.
Calculating the Discriminant First, we calculate the discriminant, which is the part under the square root: D = b 2 − 4 a c = 1 3 2 − 4 ( 3 ) ( − 7 ) = 169 + 84 = 253
Applying the Formula Now we substitute the values of a , b , c , and D into the quadratic formula to find the two possible values for x :
x = 2 ( 3 ) − 13 ± 253 = 6 − 13 ± 253 So we have two solutions: x 1 = 6 − 13 + 253 x 2 = 6 − 13 − 253
Calculating the Roots Using a calculator, we find the approximate values of x 1 and x 2 :
x 1 ≈ 6 − 13 + 15.90597 ≈ 6 2.90597 ≈ 0.4843 x 2 ≈ 6 − 13 − 15.90597 ≈ 6 − 28.90597 ≈ − 4.8177 Rounding these to two decimal places, we get: x 1 ≈ 0.48 x 2 ≈ − 4.82
Final Answer Therefore, the solutions to the equation 3 x 2 + 13 x − 7 = 0 , rounded to two decimal places, are x ≈ 0.48 and x ≈ − 4.82 .
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 predict its landing point. In business, quadratic equations can help determine the optimal price point to maximize profit, considering factors like production costs and consumer demand. Understanding how to solve these equations allows us to make informed decisions and predictions in many aspects of life.
