Which equation could be solved using this application of the quadratic formula?
[tex]x=\frac{-8 \pm \sqrt{8^2-4(3)(-2)}}{2(3)}[/tex]
A. [tex]$-2 x^2-8=10 x-3$[/tex]
B. [tex]$3 x^2-8 x-10=4$[/tex]
C. [tex]$3 x^2+8 x-10=-8$[/tex]
D. [tex]$-2 x^2+8 x-3=4$[/tex]
Answer (2)
Identify the coefficients a , b , and c from the quadratic formula: a = 3 , b = 8 , and c = − 2 .
Form the quadratic equation: 3 x 2 + 8 x − 2 = 0 .
Rearrange the given options to the standard form a x 2 + b x + c = 0 .
The equation that matches is 3 x 2 + 8 x − 10 = − 8 , which simplifies to 3 x 2 + 8 x − 2 = 0 . The answer is C.
Explanation
Understanding the Problem We are given the quadratic formula application: x = 2 ( 3 ) − 8 ± 8 2 − 4 ( 3 ) ( − 2 ) . We need to find the quadratic equation that corresponds to this formula. The general form of the quadratic formula is x = 2 a − b ± b 2 − 4 a c , which is used to solve quadratic equations of the form a x 2 + b x + c = 0 .
Identifying Coefficients Comparing the given formula x = 2 ( 3 ) − 8 ± 8 2 − 4 ( 3 ) ( − 2 ) with the general form x = 2 a − b ± b 2 − 4 a c , we can identify the coefficients: a = 3 , b = 8 , and c such that − 4 a c = − 4 ( 3 ) c = − 4 ( 3 ) ( − 2 ) . Thus, c = − 2 . So the quadratic equation is 3 x 2 + 8 x − 2 = 0 .
Checking the Options Now, let's rearrange each of the given options into the standard form a x 2 + b x + c = 0 and see which one matches 3 x 2 + 8 x − 2 = 0 .
Option A: − 2 x 2 − 8 = 10 x − 3 . Rearranging, we get − 2 x 2 − 10 x − 8 + 3 = 0 , which simplifies to − 2 x 2 − 10 x − 5 = 0 . Multiplying by -1, we get 2 x 2 + 10 x + 5 = 0 . This does not match 3 x 2 + 8 x − 2 = 0 .
Option B: 3 x 2 − 8 x − 10 = 4 . Rearranging, we get 3 x 2 − 8 x − 10 − 4 = 0 , which simplifies to 3 x 2 − 8 x − 14 = 0 . This does not match 3 x 2 + 8 x − 2 = 0 .
Option C: 3 x 2 + 8 x − 10 = − 8 . Rearranging, we get 3 x 2 + 8 x − 10 + 8 = 0 , which simplifies to 3 x 2 + 8 x − 2 = 0 . This matches our equation.
Final Answer Therefore, the correct equation is 3 x 2 + 8 x − 2 = 0 , which corresponds to option C.
Conclusion The equation that could be solved using the given application of the quadratic formula is 3 x 2 + 8 x − 10 = − 8 .
Examples
The quadratic formula is a powerful tool used in various fields, such as physics and engineering, to solve problems involving parabolic trajectories or optimizing designs. For instance, when designing a bridge, engineers use quadratic equations to model the curve of the suspension cables, ensuring structural integrity and stability. By solving these equations, they can determine the optimal cable tension and support placement, leading to a safer and more efficient design. Similarly, in projectile motion, understanding quadratic equations helps predict the range and maximum height of a projectile, crucial in fields like sports and military applications.
The equation corresponding to the given quadratic formula is 3 x 2 + 8 x − 2 = 0 . The correct multiple choice option is C, which simplifies to this equation. Therefore, Option C is the answer.
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