What is the only solution of [tex]2 x^2+8 x=x^2-16[/tex]?
Answer (2)
Rearrange the equation: 2 x 2 + 8 x = x 2 − 16 becomes x 2 + 8 x + 16 = 0 .
Factor the quadratic: x 2 + 8 x + 16 = ( x + 4 ) ( x + 4 ) .
Solve for x : ( x + 4 ) ( x + 4 ) = 0 gives x = − 4 .
The only solution is − 4 .
Explanation
Understanding the Problem We are given the equation 2 x 2 + 8 x = x 2 − 16 . Our goal is to find the only solution to this equation.
Rearranging the Equation First, let's rearrange the equation to get a quadratic equation in the standard form a x 2 + b x + c = 0 . We subtract x 2 from both sides and add 16 to both sides:
2 x 2 − x 2 + 8 x + 16 = 0
Simplifying the Equation Now, let's combine like terms to simplify the equation:
x 2 + 8 x + 16 = 0
Factoring the Quadratic Next, we factor the quadratic equation. We are looking for two numbers that multiply to 16 and add to 8 . These numbers are 4 and 4 . So, we can factor the equation as:
( x + 4 ) ( x + 4 ) = 0
Solving for x Now, we solve for x . Since both factors are the same, we have:
x + 4 = 0
Subtracting 4 from both sides, we get:
x = − 4
Verifying the Solution To verify the solution, we substitute x = − 4 into the original equation:
2 ( − 4 ) 2 + 8 ( − 4 ) = ( − 4 ) 2 − 16
2 ( 16 ) − 32 = 16 − 16
32 − 32 = 0
0 = 0
The solution is correct.
Final Answer Since the quadratic equation has a repeated root, there is only one solution. Therefore, the only solution to the equation 2 x 2 + 8 x = x 2 − 16 is x = − 4 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a ball, determining the dimensions of a garden to maximize area, or modeling the growth of a population. In physics, quadratic equations are essential for understanding projectile motion and energy calculations. For example, if you throw a ball, the height of the ball over time can be modeled by a quadratic equation, allowing you to predict when it will hit the ground. Similarly, engineers use quadratic equations to design structures and optimize their performance.
The only solution to the equation 2 x 2 + 8 x = x 2 − 16 is x = − 4 . This is found by rearranging, factoring, solving, and verifying the equation. The solution is confirmed to be correct by substitution back into the original equation.
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