Complete the square to find the solutions of $x$ for this equation.
$x^2+14 x=-13$
Select the correct answer from each drop-down menu.
The given equation is equivalent to ( $x+$ $\square$ )$^2=$ $\square$ .
The values of $x$ that make this equation true are $\square$ and $\square$ .
Answer (2)
Add ( 2 14 ) 2 = 49 to both sides of the equation: x 2 + 14 x + 49 = − 13 + 49 .
Rewrite the left side as a squared term and simplify the right side: ( x + 7 ) 2 = 36 .
Take the square root of both sides: x + 7 = ± 6 .
Solve for x : x = − 7 ± 6 , which gives x = − 1 and x = − 13 . The solutions are − 1 and − 13 .
Explanation
Problem Setup We are given the equation x 2 + 14 x = − 13 and we want to solve for x by completing the square.
Completing the Square To complete the square, we need to add a value to both sides of the equation such that the left side becomes a perfect square trinomial. The value we need to add is ( 2 14 ) 2 = 7 2 = 49 . So, we add 49 to both sides of the equation:
x 2 + 14 x + 49 = − 13 + 49
Rewriting the Equation Now, we can rewrite the left side as a squared term and simplify the right side:
( x + 7 ) 2 = 36
Taking the Square Root Next, we take the square root of both sides of the equation:
( x + 7 ) 2 = ± 36
x + 7 = ± 6
Solving for x Now, we solve for x by subtracting 7 from both sides:
x = − 7 ± 6
Finding the Solutions Finally, we find the two solutions for x :
x 1 = − 7 + 6 = − 1
x 2 = − 7 − 6 = − 13
So, the values of x that make the equation true are -1 and -13.
Examples
Completing the square is a useful technique in various real-world applications. For example, engineers use it to analyze the stability of systems, economists use it to model market behavior, and computer scientists use it in optimization algorithms. Imagine you are designing a suspension bridge and need to ensure its structural integrity. By using completing the square, you can determine the optimal cable tension to minimize stress and prevent collapse. This mathematical technique helps in making informed decisions and building safer, more efficient structures.
To solve the equation x 2 + 14 x = − 13 by completing the square, we add 49 to both sides to get ( x + 7 ) 2 = 36 . Solving this gives the solutions x = − 1 and x = − 13 .
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