26. Find the quadratic equation whose roots are
3 − √2 𝑎𝑛𝑑 3 + √2
A. 𝑥2 + 6√2𝑥 + 7 = 0
B. 𝑥2 + 6√2𝑥 − 7 = 0
C. 𝑥2 + 6𝑥 − 7 = 0
D. 𝑥2 − 6𝑥 + 7 = 0
Using the formula for finding a quadratic equation from its roots, we have:
𝑥^2 - (root1 + root2)𝑥 + root1*root2 = 0
Plugging in the given roots, we get:
𝑥^2 - (3 - √2 + 3 + √2)𝑥 + (3 - √2)(3 + √2) = 0
Simplifying, we get:
𝑥^2 - 6𝑥 + 7 = 0
Therefore, the answer is (D) 𝑥^2 − 6𝑥 + 7 = 0.
To find the quadratic equation, we need to use the fact that the roots of a quadratic equation are the values of x for which the equation equals zero.
Given the roots 3 - √2 and 3 + √2, we can write the factors as (x - (3 - √2))(x - (3 + √2)) = 0.
Expanding this equation, we have:
(x - 3 + √2)(x - 3 - √2) = 0
(x - 3)^2 - (√2)^2 = 0
(x - 3)^2 - 2 = 0
(x - 3)^2 = 2
So, the quadratic equation is 𝑥^2 - 6𝑥 + 9 - 2 = 0, which simplifies to 𝑥^2 - 6𝑥 + 7 = 0.
Therefore, the correct answer is option D. 𝑥^2 - 6𝑥 + 7 = 0.