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[tex]$-2 \sqrt{x+2}=-2 x^2-8 x-8$[/tex]
Type the correct answer in each box. Use numerals instead of words.
From least to greatest, the solutions to the equation are [tex]$x=$[/tex] $\square$ and [tex]$x=$[/tex] $\square$.
Answer (2)
Simplify the equation by dividing both sides by -2 and recognizing the perfect square.
Substitute u = x + 2 to simplify the equation to u = u 2 .
Solve for u by squaring both sides and factoring, finding u = 0 and u = 1 .
Substitute back to find x = − 2 and x = − 1 , which are the solutions to the equation: − 2 , − 1 .
Explanation
Problem Analysis We are given the equation − 2 x + 2 = − 2 x 2 − 8 x − 8 and asked to find the solutions for x .
Simplifying the Equation First, let's simplify the equation by dividing both sides by -2: x + 2 = x 2 + 4 x + 4 Notice that the right side is a perfect square: x 2 + 4 x + 4 = ( x + 2 ) 2 . So we can rewrite the equation as: x + 2 = ( x + 2 ) 2 To make this easier to solve, let's substitute u = x + 2 . Then the equation becomes: u = u 2
Solving for u Now, we square both sides of the equation to get rid of the square root: ( u ) 2 = ( u 2 ) 2 u = u 4 Rearrange the equation to set it equal to zero: u 4 − u = 0 Factor out a u :
u ( u 3 − 1 ) = 0
Finding the values of x This gives us two possible solutions for u :
u = 0
u 3 − 1 = 0 ⇒ u 3 = 1 For the first case, u = 0 . Since u = x + 2 , we have x + 2 = 0 , which means x = − 2 .
For the second case, u 3 = 1 . The real solution to this is u = 1 . Since u = x + 2 , we have x + 2 = 1 , which means x = − 1 .
Checking the solutions Now, we need to check if these solutions are valid by plugging them back into the original equation: For x = − 2 :
− 2 − 2 + 2 = − 2 ( − 2 ) 2 − 8 ( − 2 ) − 8 − 2 0 = − 2 ( 4 ) + 16 − 8 0 = − 8 + 16 − 8 0 = 0 So, x = − 2 is a valid solution. For x = − 1 :
− 2 − 1 + 2 = − 2 ( − 1 ) 2 − 8 ( − 1 ) − 8 − 2 1 = − 2 ( 1 ) + 8 − 8 − 2 = − 2 + 8 − 8 − 2 = − 2 So, x = − 1 is also a valid solution.
Final Answer The solutions are x = − 2 and x = − 1 . Since we need to list them from least to greatest, the solutions are x = − 2 and x = − 1 .
Examples
Understanding how to solve equations like this is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of an object under certain forces, you might encounter an equation involving square roots and polynomials. Solving this equation would allow you to determine the object's position at a specific time. Similarly, in electrical engineering, you might need to solve equations of this form to analyze the behavior of circuits. The ability to manipulate and solve such equations is a fundamental skill in these areas.
The solutions to the equation − 2 x + 2 = − 2 x 2 − 8 x − 8 are x = − 2 and x = − 1 . These solutions are found by simplifying the equation and using substitution, followed by verifying the solutions. The final answer is x = − 2 and x = − 1 .
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