Solve the equation by using the quadratic formula.
[tex]8 x^2+3 x=5[/tex]
a. [tex]x=\frac{8}{5},-1[/tex]
c. [tex]x=\frac{8}{5}, 1[/tex]
b. [tex]x=\frac{8}{5}, 0[/tex]
d. [tex]x=\frac{5}{-8},-1[/tex]
Please select the best answer from the choices provided
Answer (2)
We solved the quadratic equation 8 x 2 + 3 x − 5 = 0 using the quadratic formula, yielding solutions of x = 8 5 and x = − 1 . Among the provided options, option a, x = 5 8 , − 1 , is the closest match, as it includes the correct solution x = − 1 . Therefore, option a is the best choice.
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Rewrite the equation in standard form: 8 x 2 + 3 x − 5 = 0 .
Identify coefficients: a = 8 , b = 3 , c = − 5 .
Apply the quadratic formula: x = 2 ( 8 ) − 3 ± 3 2 − 4 ( 8 ) ( − 5 ) .
Simplify to find the roots: x = − 1 , 8 5 .
The solutions are x = − 1 , 8 5 .
Explanation
Problem Analysis We are given the quadratic equation 8 x 2 + 3 x = 5 . Our goal is to solve for x using the quadratic formula and choose the correct answer from the given options.
Standard Form First, we need to rewrite the equation in the standard form a x 2 + b x + c = 0 . Subtracting 5 from both sides, we get 8 x 2 + 3 x − 5 = 0
Identifying Coefficients Now, we identify the coefficients: a = 8 , b = 3 , and c = − 5 .
Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c
Substitution Plugging in the values of a , b , and c , we have x = 2 ( 8 ) − 3 ± 3 2 − 4 ( 8 ) ( − 5 )
Simplifying the Discriminant Now, we simplify the expression under the square root: 3 2 − 4 ( 8 ) ( − 5 ) = 9 + 160 = 169
Calculating the Roots So, we have x = 16 − 3 ± 169 = 16 − 3 ± 13
Finding the Roots Now, we find the two possible values for x :
x 1 = 16 − 3 + 13 = 16 10 = 8 5 and x 2 = 16 − 3 − 13 = 16 − 16 = − 1
Comparing with Options Therefore, the solutions are x = 8 5 and x = − 1 . Comparing these solutions with the given options, we see that none of the options exactly match our solutions. However, if we rewrite 8 5 as 8 5 , we can see that option d, x = − 8 5 , − 1 is incorrect. Option a, x = 5 8 , − 1 is also incorrect. However, the correct roots are x = − 1 and x = 8 5 .
Conclusion The solutions to the quadratic equation are x = 8 5 and x = − 1 . Therefore, none of the provided options are correct. However, the closest option would be one that includes -1 as a solution. Since we are forced to choose the best answer from the choices provided, and we know that x = − 1 is a correct solution, we can check which of the options includes x = − 1 . Options a and d include x = − 1 . However, option a has x = 5 8 and option d has x = − 8 5 . Since neither of these are correct, we can conclude that there might be a typo in the question, and the correct solution should be x = 8 5 and x = − 1 .
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, and modeling the growth or decay of populations. For example, if you want to build a rectangular garden with an area of 100 square meters and you know that one side is 5 meters longer than the other, you can use a quadratic equation to find the dimensions of the garden. Understanding how to solve quadratic equations is essential for solving many practical problems in engineering, physics, and economics.
