Solve $2 x^2+5 x+2=0$ by applying the quadratic formula.
Answer (2)
Identify the coefficients: a = 2 , b = 5 , c = 2 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 2 ( 2 ) − 5 ± 5 2 − 4 ( 2 ) ( 2 ) = 4 − 5 ± 9 = 4 − 5 ± 3 .
Calculate the two solutions: x 1 = 4 − 5 + 3 = − 2 1 and x 2 = 4 − 5 − 3 = − 2 . The final answer is x = − 2 1 , x = − 2 .
Explanation
Identifying Coefficients We are asked to solve the quadratic equation 2 x 2 + 5 x + 2 = 0 using the quadratic formula. Let's identify the coefficients a , b , and c in the general form of a quadratic equation a x 2 + b x + c = 0 . In our case, we have a = 2 , b = 5 , and c = 2 .
Applying the Quadratic Formula The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c We will now substitute the values of a , b , and c into this formula.
Substituting Values Substituting a = 2 , b = 5 , and c = 2 into the quadratic formula, we get: x = 2 ( 2 ) − 5 ± 5 2 − 4 ( 2 ) ( 2 ) Now, let's simplify the expression under the square root.
Simplifying the Expression We have 5 2 − 4 ( 2 ) ( 2 ) = 25 − 16 = 9 . So, the expression becomes: x = 4 − 5 ± 9 Now, we calculate the square root of 9, which is 3.
Calculating the Square Root Now we have: x = 4 − 5 ± 3 We will now find the two possible values for x .
Finding the Two Values of x The two possible values for x are: x 1 = 4 − 5 + 3 = 4 − 2 = − 2 1 x 2 = 4 − 5 − 3 = 4 − 8 = − 2 Thus, the solutions are x = − 2 1 and x = − 2 .
Final Answer Therefore, the solutions to the quadratic equation 2 x 2 + 5 x + 2 = 0 are x = − 2 1 and x = − 2 .
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. Similarly, architects use quadratic equations to design structures, ensuring stability and optimal use of space. Understanding quadratic equations helps in making informed decisions and solving practical problems in diverse fields.
To solve the equation 2 x 2 + 5 x + 2 = 0 , we use the quadratic formula. After identifying the coefficients and simplifying, we find the solutions to be x = − 2 1 and x = − 2 .
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