To find the quadratic equation given its roots, we can use the fact that if a quadratic equation has roots α and β, then the equation can be written as:
(x - α)(x - β) = 0
In this case, the roots are α = 2 + 2√5 and β = 2 - 2√5. Substituting these values into the equation, we have:
(x - (2 + 2√5))(x - (2 - 2√5)) = 0
Expanding the equation, we get:
(x - 2 - 2√5)(x - 2 + 2√5) = 0
Using the difference of squares, we can further simplify this equation:
[(x - 2) - 2√5][(x - 2) + 2√5] = 0
(x - 2)^2 - (2√5)^2 = 0
(x - 2)^2 - 4(5) = 0
(x - 2)^2 - 20 = 0
So, the quadratic equation with roots 2 ± 2√5 is (x - 2)^2 - 20 = 0.
Answers & Comments
Answer:
2 ± 2√5 is (x - 2)^2 - 20 = 0.
Step-by-step explanation:
To find the quadratic equation given its roots, we can use the fact that if a quadratic equation has roots α and β, then the equation can be written as:
(x - α)(x - β) = 0
In this case, the roots are α = 2 + 2√5 and β = 2 - 2√5. Substituting these values into the equation, we have:
(x - (2 + 2√5))(x - (2 - 2√5)) = 0
Expanding the equation, we get:
(x - 2 - 2√5)(x - 2 + 2√5) = 0
Using the difference of squares, we can further simplify this equation:
[(x - 2) - 2√5][(x - 2) + 2√5] = 0
(x - 2)^2 - (2√5)^2 = 0
(x - 2)^2 - 4(5) = 0
(x - 2)^2 - 20 = 0
So, the quadratic equation with roots 2 ± 2√5 is (x - 2)^2 - 20 = 0.