Solve $x^2=12 x-15$ by completing the square. Which is the solution set of the equation?
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\left\{-6-\sqrt{51},-6+\sqrt{51}\right\}
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\left\{-6-\sqrt{21},-6+\sqrt{21}\right\}
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\left\{6-\sqrt{51}, 6+\sqrt{51}\right\}
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\left\{6-\sqrt{21}, 6+\sqrt{21}\right\}
Answer (2)
Rewrite the equation: x 2 − 12 x = − 15 .
Complete the square: x 2 − 12 x + 36 = − 15 + 36 , which simplifies to ( x − 6 ) 2 = 21 .
Take the square root: x − 6 = ± 21 .
Solve for x : x = 6 ± 21 . The solution set is 6 − 21 , 6 + 21 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 = 12 x − 15 . Our goal is to solve this equation by completing the square and identify the correct solution set from the given options.
Rewriting the Equation First, we rewrite the equation in the form x 2 − 12 x = − 15 . This sets us up to complete the square.
Completing the Square To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. We take half of the coefficient of the x term, which is − 12 , and square it: ( 2 − 12 ) 2 = ( − 6 ) 2 = 36 . So, we add 36 to both sides of the equation: x 2 − 12 x + 36 = − 15 + 36 .
Rewriting as a Squared Term Now, we rewrite the left side as a squared term: ( x − 6 ) 2 = 21 .
Taking the Square Root Next, we take the square root of both sides of the equation: x − 6 = ± 21 .
Solving for x Now, we solve for x by adding 6 to both sides: x = 6 ± 21 . This gives us two possible solutions for x .
Identifying the Solution Set Therefore, the solution set is 6 − 21 , 6 + 21 . Comparing this with the given options, we find that the correct solution set is 6 − 21 , 6 + 21 .
Examples
Completing the square is a useful technique in various fields. For example, in physics, it can be used to find the vertex form of a projectile's trajectory equation, allowing us to easily determine the maximum height and range of the projectile. Suppose the height of a ball thrown is given by h ( t ) = − t 2 + 8 t + 2 . By completing the square, we can rewrite this as h ( t ) = − ( t − 4 ) 2 + 18 , which tells us the maximum height of the ball is 18 units and it occurs at time t = 4 .
The solutions to the equation x 2 = 12 x − 15 after completing the square are { 6 - \sqrt{21}, 6 + \sqrt{21} }. Therefore, the correct answer is { 6 - \sqrt{21}, 6 + \sqrt{21} }.
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