Find a value of [tex]$x$[/tex] if [tex]$4\left(3 x^2+2\right)=776$[/tex]
Answer (2)
Divide both sides of the equation by 4: 3 x 2 + 2 = 194 .
Subtract 2 from both sides: 3 x 2 = 192 .
Divide both sides by 3: x 2 = 64 .
Take the square root of both sides: x = ± 8 . Therefore, x = 8 or x = − 8 , so the answer is 8 .
Explanation
Understanding the Problem We are given the equation 4 ( 3 x 2 + 2 ) = 776 and we need to find a value of x that satisfies this equation.
Dividing by 4 First, divide both sides of the equation by 4 to isolate the term with x 2 :
4 4 ( 3 x 2 + 2 ) = 4 776 3 x 2 + 2 = 194
Subtracting 2 Next, subtract 2 from both sides of the equation to further isolate the x 2 term: 3 x 2 + 2 − 2 = 194 − 2 3 x 2 = 192
Dividing by 3 Now, divide both sides of the equation by 3 to isolate x 2 :
3 3 x 2 = 3 192 x 2 = 64
Taking the Square Root Finally, take the square root of both sides of the equation to solve for x :
x = ± 64 x = ± 8
Finding the Solution Thus, the possible values for x are x = 8 and x = − 8 . We can choose either value as a solution.
Examples
Imagine you are designing a square garden and need to determine the length of each side to cover a specific area. This problem is similar to finding the side length of a square ( x ) when you know that a modified area ( 3 x 2 + 2 ) scaled by a factor (4) equals a total area (776). By solving for x , you determine the dimensions of your garden, ensuring it fits your design requirements. This type of algebraic problem is useful in various real-world applications, such as construction, engineering, and design, where determining dimensions based on area or volume is essential.
To solve the equation 4 ( 3 x 2 + 2 ) = 776 , we first simplify it to 3 x 2 = 192 by dividing by 4 and then subtracting 2. We then find x 2 = 64 and take the square root, resulting in x = 8 or x = − 8 . Hence, both 8 and − 8 are possible values for x , but 8 is typically preferred as the primary solution.
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