Solve the quadratic equation by completing the square.
–3x2 + 9x = 1
(1 point)
Responses
3/2 plus or minus square root 6/6
Image with alt text: 3/2 plus or minus square root 6/6
-3 plus or minus square root 69/3
Image with alt text: -3 plus or minus square root 69/3
3 plus or minus square root -3/3
Image with alt text: 3 plus or minus square root -3/3
3 over 2 plus or minus root 69 over 6
To solve the quadratic equation by completing the square, we need to rearrange the equation to the form of (x - h)^2 = k.
Starting with -3x^2 + 9x = 1, we first move the constant term to the right side:
-3x^2 + 9x - 1 = 0
Next, we divide the entire equation by -3 to make the coefficient of x^2 equal to 1:
x^2 - 3x/(-3) + 1/(-3) = 0
Simplifying, we get:
x^2 - 3x/(-3) + 1/(-3) = x^2 + x - 1/3 = 0
Now we need to complete the square by adding and subtracting a constant term that will make the equation a perfect square trinomial. The constant term is (b/2)^2, where b is the coefficient of x.
In this case, b = 1, so (1/2)^2 = 1/4 is the constant term we need to add and subtract:
x^2 + x - 1/3 + 1/4 - 1/4 = 0
Next, we combine like terms:
x^2 + x + 1/4 - 1/3 - 1/4 = 0
Common denominator for 1/3 and 1/4 is 12:
x^2 + x + 3/12 - 4/12 - 3/12 = 0
Simplifying, we get:
x^2 + x - 1/12 = 0
Now we can rewrite the left side as a perfect square trinomial:
(x + 1/2)^2 - 1/12 = 0
To isolate (x + 1/2)^2, we add 1/12 to both sides:
(x + 1/2)^2 = 1/12
Now we can take the square root of both sides:
x + 1/2 = ± √(1/12)
To remove the fraction, we can simplify the square root:
x + 1/2 = ± √(1)/√(12) = ± √(1)/√(4*3) = ± √(1)/(2√3) = ± 1/(2√3)
Finally, we isolate x by subtracting 1/2 from both sides:
x = -1/2 ± 1/(2√3)
Simplifying the expression, we get:
x = (-1 ± √3)/(2√3)
Therefore, the correct solution is:
-1/2 plus or minus square root 3 over 2 square root 3