Fiona is completing the square to solve the polynomial equation:
[tex]x^2-4 x+1=0[/tex]
What should be her first step?
A. isolating the constant 1
B. adding 4 to both sides of the equation
C. isolating the second term, [tex]-4 x[/tex]
D. adding 1 to both sides of the equation
Answer (1)
Isolate the constant term by subtracting it from both sides of the equation.
Rewrite the equation as x 2 − 4 x = − 1 .
The first step is isolating the constant 1.
Therefore, the first step is isolating the constant 1.
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 4 x + 1 = 0 and asked to determine the first step in completing the square. Completing the square involves rewriting the quadratic expression in the form ( x + a ) 2 + b , where a and b are constants. To do this, we first isolate the constant term on one side of the equation.
Isolating the Constant Term To isolate the constant term, we subtract 1 from both sides of the equation: x 2 − 4 x + 1 − 1 = 0 − 1 x 2 − 4 x = − 1
Determining the First Step The first step in completing the square is to isolate the constant term on one side of the equation. In this case, that means isolating the constant 1.
Examples
Completing the square is a useful technique in many areas. For example, imagine you are designing a rectangular garden with a fixed area, and you want to minimize the amount of fencing needed. By completing the square, you can find the dimensions of the garden that minimize the perimeter, thus saving on fencing costs. This technique is also used in physics to solve problems involving projectile motion and optimization problems in economics.
