Solve $x^2-8 x-9=0$
Rewrite the equation so that it is of the form $x^2+b x=c$.
Answer (2)
Rewrite the equation in the form x 2 + b x = c : x 2 − 8 x = 9 .
Complete the square by adding ( 2 b ) 2 = 16 to both sides: x 2 − 8 x + 16 = 25 .
Rewrite the left side as a squared term and take the square root of both sides: ( x − 4 ) 2 = 25 , so x − 4 = ± 5 .
Solve for x : x = 9 and x = − 1 , thus the solutions are x = 9 , x = − 1 .
Explanation
Problem Setup We are given the quadratic equation x 2 − 8 x − 9 = 0 . Our goal is to solve this equation by completing the square.
Rewriting the Equation First, we rewrite the equation in the form x 2 + b x = c . To do this, we add 9 to both sides of the equation: x 2 − 8 x = 9
Completing the Square Now, we complete the square. To do this, we need to add ( 2 b ) 2 to both sides of the equation. In our case, b = − 8 , so we need to add ( 2 − 8 ) 2 = ( − 4 ) 2 = 16 to both sides: x 2 − 8 x + 16 = 9 + 16
Simplifying the Equation Simplify both sides of the equation: x 2 − 8 x + 16 = 25
Factoring the Left Side Rewrite the left side as a squared term: ( x − 4 ) 2 = 25
Taking the Square Root Take the square root of both sides: x − 4 = ± 25 x − 4 = ± 5
Solving for x Solve for x in both cases: Case 1: x − 4 = 5 x = 4 + 5 = 9 Case 2: x − 4 = − 5 x = 4 − 5 = − 1 So the solutions are x = 9 and x = − 1 .
Final Answer Therefore, the solutions to the quadratic equation x 2 − 8 x − 9 = 0 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 find the minimum value of the function f ( x ) = x 2 − 4 x + 7 . By completing the square, you can rewrite the function as f ( x ) = ( x − 2 ) 2 + 3 . Since ( x − 2 ) 2 is always non-negative, the minimum value of f ( x ) occurs when ( x − 2 ) 2 = 0 , which is when x = 2 . The minimum value is then f ( 2 ) = 3 .
To rewrite the equation x 2 − 8 x − 9 = 0 in the form x 2 + b x = c , we can move the constant to obtain x 2 − 8 x = 9 . Solving this equation through completing the square, we find the solutions are x = 9 and x = − 1 .
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