Solve the equation $x^2-16 x+54=0$ by completing the square. Fill in the values of $a$ and $b$ to complete the solutions.

Rewrite the equation: x 2 − 16 x = − 54 .
Complete the square: x 2 − 16 x + 64 = − 54 + 64 , which simplifies to ( x − 8 ) 2 = 10 .
Take the square root: x − 8 = ± 10 ​ .
Solve for x : x = 8 ± 10 ​ , thus a = 10 . The solutions are x = 8 + 10 ​ and x = 8 − 10 ​ .
The value of a is 10 ​ .

Explanation

Understanding the Problem We are given the quadratic equation x 2 − 16 x + 54 = 0 . Our goal is to solve this equation by completing the square and express the solution in the form x = 8 + a ​ and x = 8 − a ​ , where a is a constant to be determined.

Isolating the x terms First, we rewrite the equation as x 2 − 16 x = − 54 .

Completing the Square To complete the square, we need to add ( 2 − 16 ​ ) 2 = ( − 8 ) 2 = 64 to both sides of the equation. This gives us x 2 − 16 x + 64 = − 54 + 64 .

Rewriting as a Squared Term Now, we rewrite the left side as a squared term: ( x − 8 ) 2 = 10 .

Taking the Square Root Next, we take the square root of both sides: x − 8 = ± 10 ​ .

Solving for x Now, we solve for x : x = 8 ± 10 ​ .

Final Answer Finally, we express the solutions as x = 8 + 10 ​ and x = 8 − 10 ​ . Therefore, a = 10 .


Examples
Completing the square is a useful technique in physics, especially when dealing with projectile motion. For example, if you have an equation that describes the height of a ball thrown in the air as a function of time, completing the square can help you find the maximum height the ball reaches and the time at which it reaches that height. This method transforms the equation into a form that reveals the vertex of the parabola, which represents the maximum or minimum value of the function.

Answered by GinnyAnswer | 2025-07-03

To solve the equation x 2 − 16 x + 54 = 0 , we completed the square, resulting in the solutions x = 8 + 10 ​ and x = 8 − 10 ​ . The value of a is 10 .
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Answered by Anonymous | 2025-07-04