Determine which equation has the same solutions as the given equation.
[tex]x^2-10 x-11=0[/tex]
A. [tex](x-5)^2=36[/tex]
B. [tex](x-5)^2=21[/tex]
C. [tex](x-10)^2=36[/tex]
D. [tex](x-10)^2=21[/tex]
Answer (2)
Rewrite the given equation x 2 − 10 x − 11 = 0 as x 2 − 10 x = 11 .
Complete the square by adding ( 2 − 10 ) 2 = 25 to both sides: x 2 − 10 x + 25 = 11 + 25 .
Simplify the equation to ( x − 5 ) 2 = 36 .
The equation with the same solutions is ( x − 5 ) 2 = 36 .
Explanation
Analyzing the Problem We are given the quadratic equation x 2 − 10 x − 11 = 0 and we need to find an equivalent equation among the given options. The key idea here is to complete the square to rewrite the given equation in vertex form, which will allow us to easily compare it with the options.
Isolating the x terms First, we rewrite the equation as x 2 − 10 x = 11 .
Completing the Square To complete the square, we need to add ( 2 − 10 ) 2 = ( − 5 ) 2 = 25 to both sides of the equation. This gives us x 2 − 10 x + 25 = 11 + 25 .
Simplifying the Equation Now, we simplify both sides of the equation. The left side becomes a perfect square, and the right side simplifies to 36 . So, we have ( x − 5 ) 2 = 36 .
Finding the Correct Option Comparing this equation with the given options, we see that option A, ( x − 5 ) 2 = 36 , matches our result. Therefore, the equation with the same solutions as the given equation is ( x − 5 ) 2 = 36 .
Examples
Completing the square is a useful technique in many areas, such as finding the vertex of a parabola, solving quadratic equations, and even in calculus when dealing with integrals. For example, if you are designing a parabolic mirror, knowing the vertex form helps you determine the optimal placement of the light source to maximize reflection. Similarly, in physics, understanding projectile motion often involves completing the square to find the maximum height reached by a projectile.
The equation x 2 − 10 x − 11 = 0 can be rewritten as ( x − 5 ) 2 = 36 by completing the square. Comparing this with the options given, the correct answer is option A: ( x − 5 ) 2 = 36 .
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