3x^2-12x-24=0
3x^2- 12x - 24 = 0
Solve the quadratic equation by completing the square:
Divide both sides by 3:
x^2 - 4x - 8 = 0
Add 8 to both sides:
x^2 - 4x = 8
Add 4 to both sides:
x^2 - 4x + 4 = 12
Factor the left hand side:
(x - 2)^2 = 12
Take the square root of both sides:
|x - 2| = 2 sqrt(3)
Eliminate the absolute value:
x - 2 = -2 sqrt(3) or x - 2 = 2 sqrt(3)
Add 2 to both sides:
x = 2 - 2 sqrt(3) or x - 2 = 2 sqrt(3)
Add 2 to both sides:
x = 2 - 2 sqrt(3) or x = 2 + 2 sqrt(3)
A wheel made 672 revolutions at 35 1/5 revolutions per minute. How many minutes did it rotates?
To solve the equation 3x^2 - 12x - 24 = 0, we can use the quadratic formula. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 3, b = -12, and c = -24. Substitute these values into the quadratic formula:
x = (-(-12) ± √((-12)^2 - 4(3)(-24))) / (2(3))
Simplifying further:
x = (12 ± √(144 + 288)) / 6
x = (12 ± √432) / 6
Now, let's simplify the square root:
x = (12 ± √(16 * 27)) / 6
x = (12 ± 4√3) / 6
Divide both the numerator and denominator by their greatest common divisor, which is 4:
x = (3 ± √3) / 3
Therefore, the two solutions to the equation 3x^2 - 12x - 24 = 0 are:
x = (3 + √3) / 3
x = (3 - √3) / 3
To solve the equation 3x^2 - 12x - 24 = 0, we can use the quadratic formula or factoring method.
1. Using the quadratic formula:
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 3, b = -12, and c = -24. Plugging these values into the quadratic formula, we have:
x = (-(-12) ± √((-12)^2 - 4 * 3 * -24)) / (2 * 3)
x = (12 ± √(144 + 288)) / 6
x = (12 ± √432) / 6
x = (12 ± 2√108) / 6
x = (12 ± 2√(36 * 3)) / 6
x = (12 ± 2 * 6√3) / 6
x = (12 ± 12√3) / 6
x = 2 ± 2√3
So the solutions to the equation 3x^2 - 12x - 24 = 0 are:
x = 2 + 2√3
x = 2 - 2√3
2. Using factoring:
To factor the quadratic equation 3x^2 - 12x - 24 = 0, we need to find two numbers that multiply to give -24 and add up to -12 (the coefficient of x). In this case, these numbers are -6 and 4.
We can rewrite the equation as:
3x^2 - 6x + 4x - 24 = 0
Now, we can group the terms:
(3x^2 - 6x) + (4x - 24) = 0
Factor out the common terms:
3x(x - 2) + 4(x - 6) = 0
Now we have a common factor in both terms, (x - 2). We can factor it out:
(x - 2)(3x + 4) = 0
Setting each factor equal to zero, we have:
x - 2 = 0 or 3x + 4 = 0
Solving these equations gives us:
x = 2 or x = -4/3
So the solutions to the equation 3x^2 - 12x - 24 = 0 are:
x = 2
x = -4/3