What are the values of $x$ in the equation $x^2-6 x+9=25$?
A. $x=-2$ or $x=8$
B. $x=-1$ or $x=-11$
C. $x=1$ or $x=11$
D. $x=2$ or $x=-8$
Answer (2)
Rewrite the equation as a perfect square: ( x − 3 ) 2 = 25 .
Take the square root of both sides: x − 3 = ± 5 .
Solve for x in both cases: x = 3 + 5 and x = 3 − 5 .
The solutions are x = 8 and x = − 2 , so the final answer is x = − 2 or x = 8 .
Explanation
Understanding the Problem We are given the equation x 2 − 6 x + 9 = 25 . Our goal is to find the values of x that satisfy this equation.
Rewriting the Equation Notice that the left side of the equation is a perfect square trinomial. We can rewrite it as ( x − 3 ) 2 = 25 .
Taking the Square Root Now, we take the square root of both sides of the equation: ( x − 3 ) 2 = ± 25 . This simplifies to x − 3 = ± 5 .
Two Cases to Solve We now have two separate equations to solve for x :
x − 3 = 5
x − 3 = − 5
Solving for x Solving the first equation, we add 3 to both sides: x = 5 + 3 = 8 .
Solving the second equation, we add 3 to both sides: x = − 5 + 3 = − 2 .
Final Answer Therefore, the values of x that satisfy the equation are x = 8 and x = − 2 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For example, when calculating the trajectory of a projectile, you need to solve a quadratic equation to determine its range or maximum height. Similarly, in electrical engineering, quadratic equations are used to analyze circuits and determine the values of components.
The values of x that satisfy the equation x 2 − 6 x + 9 = 25 are x = 8 and x = − 2 . This can be found by rearranging the equation into standard form and applying the quadratic formula. The correct multiple choice answer is A : x = − 2 or x = 8 .
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