Solve the quadratic equation [tex]x^2-12 x+23=0[/tex] by completing the square.
A) [tex]x=6 \pm \sqrt{13}[/tex]
B) [tex]x=13 \pm \sqrt{6}[/tex]
C) [tex]x=13-\sqrt{6}[/tex]
D) [tex]x=-6 \pm \sqrt{13}[/tex]
Answer (1)
Rewrite the equation: x 2 − 12 x = − 23 .
Complete the square: Add ( 2 − 12 ) 2 = 36 to both sides: x 2 − 12 x + 36 = − 23 + 36 .
Rewrite as a squared term: ( x − 6 ) 2 = 13 .
Solve for x : x = 6 ± 13 . The final answer is x = 6 \tpm 13 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 12 x + 23 = 0 and asked to solve it by completing the square. This method involves rewriting the quadratic expression in the form ( x − h ) 2 + k = 0 , which allows us to easily solve for x .
Isolating the x terms First, we rewrite the equation as x 2 − 12 x = − 23 .
Calculating the Constant 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. We take half of the coefficient of the x term, which is − 12 , and square it: ( 2 − 12 ) 2 = ( − 6 ) 2 = 36 .
Adding the Constant to Both Sides Now, we add 36 to both sides of the equation: x 2 − 12 x + 36 = − 23 + 36 .
Rewriting as a Squared Term We can rewrite the left side as a squared term: ( x − 6 ) 2 = 13 .
Taking the Square Root Next, we take the square root of both sides: x − 6 = ± 13 .
Solving for x Finally, we solve for x : x = 6 ± 13 .
Final Answer Therefore, the solutions to the quadratic equation are x = 6 + 13 and x = 6 − 13 . The correct answer is A) x = 6 \tpm 13
Examples
Completing the square is a useful technique in various real-world scenarios. For example, engineers use it to optimize designs, such as finding the dimensions of a rectangular enclosure that maximizes the enclosed area for a given perimeter. Similarly, in physics, it can be used to determine the maximum height reached by a projectile or to analyze the behavior of oscillating systems. This method provides a structured way to solve quadratic equations, which appear frequently in modeling physical phenomena.
