Solve for x.
$x^2-2 x-19=0$

Identify the coefficients: a = 1 , b = − 2 , and c = − 19 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ .
Substitute the values and simplify: x = 2 2 ± 80 ​ ​ = 2 2 ± 4 5 ​ ​ .
Obtain the solutions: x = 1 ± 2 5 ​ .

The solutions to the quadratic equation are x = 1 ± 2 5 ​ ​ .
Explanation

Problem Analysis We are given the quadratic equation x 2 − 2 x − 19 = 0 . Our goal is to find the values of x that satisfy this equation.

Quadratic Formula To solve the quadratic equation, we will use the quadratic formula, which is given by: x = 2 a − b ± b 2 − 4 a c ​ ​ where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 .

Substituting Values In our equation, x 2 − 2 x − 19 = 0 , we have a = 1 , b = − 2 , and c = − 19 . Substituting these values into the quadratic formula, we get: x = 2 ( 1 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 1 ) ( − 19 ) ​ ​

Simplifying the Expression Now, we simplify the expression: x = 2 2 ± 4 + 76 ​ ​ = 2 2 ± 80 ​ ​ Since 80 ​ = 16 × 5 ​ = 4 5 ​ , we can further simplify: x = 2 2 ± 4 5 ​ ​ Dividing both terms in the numerator by 2, we get: x = 1 ± 2 5 ​ Thus, the two solutions for x are 1 + 2 5 ​ and 1 − 2 5 ​ .

Final Answer Therefore, the solutions to the quadratic equation x 2 − 2 x − 19 = 0 are x = 1 + 2 5 ​ and x = 1 − 2 5 ​ .


Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 100 square meters and you know that the length must be 5 meters longer than the width, you can set up a quadratic equation to find the dimensions of the garden. Understanding how to solve quadratic equations allows you to solve these types of practical problems.

Answered by GinnyAnswer | 2025-07-07

To solve the equation x 2 − 2 x − 19 = 0 , we use the quadratic formula to find the solutions. The solutions are x = 1 + 2 5 ​ and x = 1 − 2 5 ​ . This method helps identify values of x that satisfy the equation.
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Answered by Anonymous | 2025-07-21