Solve $x^2=12 x-15$ by completing the square. Which is the solution set of the equation?
$\left\{-6-\sqrt{51},-6+\sqrt{51}\right\}$
$\left\{-6-\sqrt{21},-6+\sqrt{21}\right\}$
$\left\{6-\sqrt{51}, 6+\sqrt{51}\right\}$
$\left\{6-\sqrt{21}, 6+\sqrt{21}\right\}$
Answer (2)
Rewrite the equation as x 2 − 12 x = − 15 .
Complete the square by adding ( 2 − 12 ) 2 = 36 to both sides: x 2 − 12 x + 36 = − 15 + 36 .
Rewrite the left side as a squared term: ( x − 6 ) 2 = 21 .
Solve for x : x = 6 ± 21 , so the solution set is 6 − 21 , 6 + 21 .
Explanation
Rewrite the equation First, we need to rewrite the given equation x 2 = 12 x − 15 in the standard form for completing the square.
Isolate the quadratic and linear terms Subtract 12 x from both sides to get x 2 − 12 x = − 15 .
Calculate the constant to complete the square To complete the square, we need to add a constant to both sides of the equation such that the left side becomes a perfect square. The constant is ( 2 b ) 2 , where b is the coefficient of the x term. In this case, b = − 12 , so the constant is ( 2 − 12 ) 2 = ( − 6 ) 2 = 36 .
Add the constant to both sides Add 36 to both sides of the equation: x 2 − 12 x + 36 = − 15 + 36 .
Rewrite as a perfect square Now, rewrite the left side as a squared term: ( x − 6 ) 2 = 21 .
Take the square root Take the square root of both sides: x − 6 = ± 21 .
Solve for x Solve for x by adding 6 to both sides: x = 6 ± 21 .
State the solution set Therefore, the solution set is 6 − 21 , 6 + 21 .
Examples
Completing the square is a useful technique in physics, especially when dealing with projectile motion. For instance, if you want to find the maximum height of a projectile, you often end up with a quadratic equation representing the height as a function of time. By completing the square, you can easily rewrite the equation in vertex form, which directly gives you the maximum height and the time at which it occurs. This method allows physicists and engineers to optimize designs and predict outcomes in various real-world scenarios, such as designing efficient ballistics or optimizing the trajectory of a rocket.
By completing the square for the equation x 2 = 12 x − 15 , we find the solution set as { 6 − 21 , 6 + 21 } . This involves rewriting the equation, isolating terms, and calculating the necessary constant to complete the square. Ultimately, solving leads us to these two results for x .
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