Question

Solve for x. 3x2−6x+2=0

x=3±23√3
x=3±3√3 x equals fraction numerator 3 plus or minus square root 3 end root end numerator over 3 end fraction x=6±43√3 x equals fraction numerator 6 plus or minus 4 square root 3 end root end numerator over 3 end fraction x=6±23√3 x equals fraction numerator 6 plus or minus 2 square root 3 end root end numerator over 3 end fraction

Answer 1

Using the quadratic formula, we solve the equation 3x2-6x+2=0 and find the solutions to be x=3√3/3 or x=−2√3/3.

Step-by-step explanation:

To solve the quadratic equation 3x2−6x+2=0, we use the quadratic formula, which is x = −b ± √(b2 − 4ac) / (2a). In this case, a = 3, b = −6, and c = 2. Plugging these values into the formula, we calculate the determinant (Δ) as:

Δ = b2 − 4ac = (−6)2 − 4(3)(2) = 36 − 24 = 12

Since the determinant is positive, we have two real and distinct roots. Continuing with the quadratic formula:

x = (6 ± √12) / (2 × 3)

Which simplifies to:

x = 1 ± √(12/9)

x = 1 ± √(4/3)

x = 1 ± √4/√3

x = 1 ± 2/√3

Finally, to rationalize the denominator:

x = (1 ± 2/√3) × (√3/√3)

x = (√3 ± 2√3) / 3

Therefore, the solutions for x are:

x = (√3 + 2√3) / 3

x = (√3 − 2√3) / 3

x = 3√3 / 3 or −2√3 / 3