Which is a solution to $(x-3)(x+9)=-27$?
A. $x = -9$
B. $x = -3$
C. $x = 0$
D. $x = 6$
Answer (2)
Expand the left side of the equation: ( x − 3 ) ( x + 9 ) = x 2 + 6 x − 27 .
Rewrite the equation: x 2 + 6 x − 27 = − 27 , which simplifies to x 2 + 6 x = 0 .
Factor the equation: x ( x + 6 ) = 0 .
Solve for x : x = 0 or x = − 6 . Comparing with the given options, the solution is 0 .
Explanation
Understanding the Problem We are given the equation ( x − 3 ) ( x + 9 ) = − 27 and asked to find which of the given values, x = − 9 , − 3 , 0 , 6 , is a solution.
Expanding the Equation First, let's expand the left side of the equation: ( x − 3 ) ( x + 9 ) = x 2 + 9 x − 3 x − 27 = x 2 + 6 x − 27
Rewriting the Equation Now, we rewrite the equation as: x 2 + 6 x − 27 = − 27
Simplifying the Equation Next, we simplify the equation by adding 27 to both sides: x 2 + 6 x = 0
Factoring the Equation Now, we factor the left side of the equation: x ( x + 6 ) = 0
Solving for x Solving for x , we get two possible solutions: x = 0 or x = − 6
Checking the Options We are given the options x = − 9 , − 3 , 0 , 6 . Comparing our solutions with the given options, we see that x = 0 is one of the given options.
Final Answer Therefore, x = 0 is a solution to the equation ( x − 3 ) ( x + 9 ) = − 27 .
Examples
Understanding quadratic equations and their solutions is crucial in many fields, such as physics and engineering. For instance, when designing a bridge, engineers need to solve quadratic equations to calculate the optimal shape and load distribution to ensure the bridge's stability. Similarly, in physics, projectile motion can be modeled using quadratic equations, allowing us to predict the trajectory of objects, like a ball thrown in the air. These equations help us understand and predict real-world phenomena.
The equation ( x − 3 ) ( x + 9 ) = − 27 can be solved by expanding and simplifying it to find possible values for x . The solutions we found are x = 0 and x = − 6 , and among the multiple-choice options given, x = 0 is the correct answer. Thus, the solution is C : x = 0 .
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