Solve for $x$ in the equation $x^2+20 x+100=36$

Rewrite the equation: x 2 + 20 x + 100 = 36 becomes x 2 + 20 x + 64 = 0 .
Factor the quadratic: x 2 + 20 x + 64 = ( x + 4 ) ( x + 16 ) .
Set each factor to zero: x + 4 = 0 or x + 16 = 0 .
Solve for x : x = − 4 or x = − 16 . Thus, the solutions are x = − 16 or x = − 4 ​ .

Explanation

Problem Analysis We are given the equation x 2 + 20 x + 100 = 36 . Our goal is to solve for x .

Rewrite the Equation First, we rewrite the equation by subtracting 36 from both sides to set the equation to zero: x 2 + 20 x + 100 − 36 = 0

Simplify Simplify the equation: x 2 + 20 x + 64 = 0

Factor the Quadratic Now, we need to factor the quadratic expression x 2 + 20 x + 64 . We are looking for two numbers that multiply to 64 and add up to 20. These numbers are 4 and 16. So, we can factor the expression as: ( x + 4 ) ( x + 16 ) = 0

Solve for x To solve for x , we set each factor equal to zero: x + 4 = 0 or x + 16 = 0

Final Values of x Solving these equations gives us the two possible values for x :
x = − 4 or x = − 16

Conclusion Therefore, the solutions for x are x = − 4 and x = − 16 .


Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a ball, determining the dimensions of a garden to maximize area, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 64 square feet and a perimeter that requires 40 feet of fencing, you can use a quadratic equation to find the dimensions of the garden. The equation x 2 − 20 x + 64 = 0 models this situation, where x represents the length of one side of the garden. Solving this equation gives you the possible lengths for the garden sides, helping you plan your garden efficiently.

Answered by GinnyAnswer | 2025-07-03

The solutions for the equation x 2 + 20 x + 100 = 36 are x = − 4 and x = − 16 .
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Answered by Anonymous | 2025-07-04