Solve the quadratic equation:
[tex]x^2-2 x+6=0[/tex]

A. [tex]x=2 \pm \sqrt{20}[/tex]
B. [tex]x=2 \pm \sqrt{28}[/tex]
C. [tex]x=-2 \pm \sqrt{28}[/tex]
D. no real solution

Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c ​ ​ with a = 1 , b = − 2 , and c = 6 .
Substitute the values into the formula: x = 2 ( 1 ) 2 ± ( − 2 ) 2 − 4 ( 1 ) ( 6 ) ​ ​ .
Simplify to find x = 2 2 ± − 20 ​ ​ .
Since the discriminant is negative, there are no real solutions: no real solution ​ .

Explanation

Problem Analysis We are given the quadratic equation x 2 − 2 x + 6 = 0 and asked to find its solution.

Applying Quadratic Formula We can use the quadratic formula to solve this equation. The quadratic formula 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 . In this case, a = 1 , b = − 2 , and c = 6 .

Substitution Substitute the values of a , b , and c into the quadratic formula:


x = 2 ( 1 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 1 ) ( 6 ) ​ ​

Simplification Simplify the expression:

x = 2 2 ± 4 − 24 ​ ​
x = 2 2 ± − 20 ​ ​

Complex Solutions Since the discriminant (the value inside the square root) is negative, the solutions are complex numbers. We can rewrite − 20 ​ as 20 ​ i , where i is the imaginary unit ( i 2 = − 1 ). Thus,

x = 2 2 ± 20 ​ i ​

Final Simplification and Conclusion Further simplify:

x = 1 ± 2 20 ​ ​ i
x = 1 ± 2 4 × 5 ​ ​ i
x = 1 ± 2 2 5 ​ ​ i
x = 1 ± 5 ​ i
Since the question asks for real solutions and we found complex solutions, there are no real solutions.
Examples
Quadratic equations appear in various real-world scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area with a specific perimeter and area, or modeling population growth. For instance, if you're launching a rocket, the height of the rocket over time can be modeled by a quadratic equation, where solving the equation helps determine when the rocket will hit the ground. Understanding quadratic equations is essential for solving many practical problems in physics, engineering, and other fields.

Answered by GinnyAnswer | 2025-07-08

The quadratic equation x 2 − 2 x + 6 = 0 has no real solutions because the discriminant is negative. The solutions are complex, indicated by the presence of the imaginary unit i . Therefore, the correct option is D: no real solution.
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Answered by Anonymous | 2025-07-16