Use completing the square to solve for $x$ in the equation $(x-12)(x+4)=9$.
A. $x=-1$ or 15
B. $x=1$ or 7
C. $x=4 \pm \sqrt{41}$
D. $x=4 \pm \sqrt{73}$
Answer (2)
Expand the original equation to get x 2 − 8 x − 48 = 9 .
Isolate the x terms: x 2 − 8 x = 57 .
Complete the square by adding ( − 8/2 ) 2 = 16 to both sides: ( x − 4 ) 2 = 73 .
Solve for x : x = 4 ± 73 .
Explanation
Expand the equation We are given the equation ( x − 12 ) ( x + 4 ) = 9 and asked to solve for x by completing the square. First, we need to expand the left side of the equation.
Rewrite the equation Expanding the left side, we have: ( x − 12 ) ( x + 4 ) = x 2 + 4 x − 12 x − 48 = x 2 − 8 x − 48 So the equation becomes: x 2 − 8 x − 48 = 9
Isolate x terms Next, we want to isolate the x 2 and x terms on one side of the equation. Add 48 to both sides: x 2 − 8 x = 9 + 48 x 2 − 8 x = 57
Complete the square Now, we complete the square. To do this, we take half of the coefficient of the x term, which is − 8 , and square it. Half of − 8 is − 4 , and ( − 4 ) 2 = 16 . We add 16 to both sides of the equation: x 2 − 8 x + 16 = 57 + 16 x 2 − 8 x + 16 = 73
Rewrite as a square The left side is now a perfect square, so we can rewrite it as: ( x − 4 ) 2 = 73
Take the square root Take the square root of both sides: x − 4 = ± 73
Solve for x Finally, solve for x by adding 4 to both sides: x = 4 ± 73
Final Answer Therefore, the solutions are x = 4 + 73 and x = 4 − 73 .
Examples
Completing the square is a useful technique in physics, especially when dealing with projectile motion. For example, if you want to find the maximum height of a ball thrown upwards, you can model the height as a quadratic equation and complete the square to find the vertex, which represents the maximum height. This method helps in optimizing various physical scenarios, like determining the ideal launch angle for maximum range or height.
We solved the equation ( x − 12 ) ( x + 4 ) = 9 by expanding, isolating the x terms, and completing the square. This led us to the perfect square form ( x − 4 ) 2 = 73 , resulting in the solutions x = 4 ± 73 . Therefore, the correct answer is option D: x = 4 ± 73 .
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