use the quadratic formula to solve the equation.
3x^2 - 41x = -110
First, rewrite the equation in standard form:
3x^2 - 41x + 110 = 0
Now, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in the values:
a = 3
b = -41
c = 110
x = (-(-41) ± sqrt((-41)^2 - 4(3)(110))) / 2(3)
x = (41 ± sqrt(41^2 - 1320)) / 6
Simplifying:
x = (41 ± sqrt(401)) / 6
x = (41 ± 20.02) / 6
x = 10.67 or x = 5.33
Therefore, the solutions to the equation 3x^2 - 41x = -110 are x = 10.67 and x = 5.33.
To solve the equation using the quadratic formula, first let's write the equation in standard form:
3x^2 - 41x + 110 = 0
Now let's identify the values needed for the quadratic formula:
a = 3
b = -41
c = 110
The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
x = (41 ± √((-41)^2 - 4 * 3 * 110)) / (2 * 3)
Simplifying further:
x = (41 ± √(1681 - 1320)) / 6
x = (41 ± √(361)) / 6
Now, we can calculate the solutions:
x1 = (41 + √(361)) / 6
x2 = (41 - √(361)) / 6
Let's calculate the values of x1 and x2:
x1 = (41 + 19) / 6
x1 = 60 / 6
x1 = 10
x2 = (41 - 19) / 6
x2 = 22 / 6
x2 ≈ 3.67
Therefore, the solutions to the equation 3x^2 - 41x = -110 are x = 10 and x ≈ 3.67.
To solve the quadratic equation 3x^2 - 41x = -110 using the quadratic formula, we first need to rearrange the equation in the standard quadratic form, which is ax^2 + bx + c = 0. Comparing it with the given equation, we have:
3x^2 - 41x + 110 = 0
Now, let's identify the values for a, b, and c:
a = 3
b = -41
c = 110
The quadratic formula states that the solutions for a quadratic equation of the form ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values we obtained, we have:
x = (-(41) ± √((-41)^2 - 4(3)(110))) / (2(3))
Now, let's calculate the discriminant:
D = (b^2 - 4ac)
D = (-41)^2 - 4(3)(110)
D = 1681 - 1320
D = 361
Since the discriminant (D) is positive, there are two distinct real solutions.
Applying the quadratic formula:
x = (-(41) ± √(361)) / (2(3))
x = (-(41) ± √361) / 6
x = (-(41) ± 19) / 6
Now we can find the two solutions for x:
x₁ = (-(41) + 19) / 6
x₁ = (-22) / 6
x₁ = -11/3
x₂ = (-(41) - 19) / 6
x₂ = (-60) / 6
x₂ = -10
Therefore, the solutions to the quadratic equation 3x^2 - 41x = -110 are x = -11/3 and x = -10.