Solve $x^2-8 x-9=0$
Rewrite the equation so that it is of the form $x^2+b x=c$.
$x^2-8x=9$
Add 16 to each side to complete the square.
$x^2-8 x+16=9+16$
Now that you have $x^2-8 x+16=9+16$, apply the square root property to the equation.
Answer (1)
Rewrite the equation: x 2 − 8 x = 9 .
Complete the square: x 2 − 8 x + 16 = 9 + 16 , which simplifies to ( x − 4 ) 2 = 25 .
Apply the square root property: x − 4 = ± 5 .
Solve for x : x = 4 + 5 = 9 and x = 4 − 5 = − 1 . The solutions are x = 9 , − 1 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 8 x − 9 = 0 and we need to solve it by completing the square and applying the square root property.
Rewriting the Equation First, we rewrite the equation in the form x 2 + b x = c . Adding 9 to both sides of the equation, we get: x 2 − 8 x = 9
Completing the Square Next, we complete the square. To do this, we take half of the coefficient of the x term, which is − 8 , and square it: ( 2 − 8 ) 2 = ( − 4 ) 2 = 16 . We add this value to both sides of the equation: x 2 − 8 x + 16 = 9 + 16
Simplifying the Equation Now, we simplify both sides of the equation. The left side is a perfect square, and the right side is a simple sum: ( x − 4 ) 2 = 25
Applying the Square Root Property We apply the square root property by taking the square root of both sides: x − 4 = ± 25 x − 4 = ± 5
Solving for x Finally, we solve for x by adding 4 to both sides: x = 4 ± 5 This gives us two possible solutions: x 1 = 4 + 5 = 9 x 2 = 4 − 5 = − 1 Thus, the solutions are x = 9 and x = − 1 .
Examples
Completing the square is a useful technique in many areas, such as finding the vertex of a parabola or solving optimization problems. For example, suppose you want to fence off a rectangular garden with a fixed perimeter. Completing the square can help you determine the dimensions that maximize the garden's area. This technique is also used in physics to analyze oscillatory motion and in engineering to design stable systems.
