Dimitri is solving the equation $x^2-10 x=21$. Which value must be added to both sides of the equation to make the left side a perfect-square trinomial?
A. 0
B. 4
C. 25
D. 100
Answer (2)
Identify the coefficient of the x term, which is -10.
Divide the coefficient by 2: 2 − 10 = − 5 .
Square the result: ( − 5 ) 2 = 25 .
Add 25 to both sides of the equation to complete the square: 25 .
Explanation
Understanding the Problem We are given the equation x 2 − 10 x = 21 and asked to find the value that must be added to both sides to make the left side a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial.
Completing the Square To complete the square for the quadratic expression x 2 − 10 x , we need to add a value to both sides of the equation such that the left side becomes a perfect square trinomial. The general form of a perfect square trinomial is x 2 + b x + c , where c = ( 2 b ) 2 . In our case, b = − 10 .
Calculating the Value We calculate the value to be added as follows: c = ( 2 b ) 2 = ( 2 − 10 ) 2 = ( − 5 ) 2 = 25
Adding to Both Sides Therefore, we must add 25 to both sides of the equation to make the left side a perfect square trinomial. The new equation becomes: x 2 − 10 x + 25 = 21 + 25 ( x − 5 ) 2 = 46
Final Answer The value that must be added to both sides of the equation to make the left side a perfect square trinomial is 25.
Examples
Completing the square is a useful technique in many areas of mathematics and physics. For example, in physics, it can be used to find the minimum potential energy of a system. In engineering, it can be used to optimize the design of a system. In real life, completing the square can be used to solve optimization problems, such as finding the dimensions of a garden that maximize the area for a given perimeter. It also helps in understanding the vertex form of a parabola, which is useful in analyzing projectile motion.
To make the left side of the equation x 2 − 10 x = 21 a perfect-square trinomial, we must add 25 to both sides. When we add 25, the equation becomes ( x − 5 ) 2 = 46 . Therefore, the answer is 25.
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