identify the intervals on which the quadratic function is positive
y=2x^2 -15x+18
To determine the intervals on which the quadratic function is positive, we need to find the x-values for which the function's y-values are greater than zero.
To do this, we can set the quadratic function equal to zero and solve for x:
2x^2 - 15x + 18 = 0
Factoring the quadratic equation, we have:
(2x - 3)(x - 6) = 0
Setting each factor equal to zero gives us:
2x - 3 = 0 or x - 6 = 0
Solving these equations, we find:
2x = 3 -> x = 3/2
x = 6
Now we have three critical values: x = 3/2, x = 6, and the vertex of the quadratic function.
We know that the quadratic function is positive when the value of y is greater than zero.
To find the intervals on which the quadratic function is positive, we only need to consider the intervals between the critical values.
In this case, we have the following intervals:
(-∞, 3/2), (3/2, 6), (6, ∞)
Therefore, the quadratic function y = 2x^2 - 15x + 18 is positive on the intervals (-∞, 3/2) and (6, ∞).
To determine the intervals on which the quadratic function y=2x^2 -15x+18 is positive, we can follow these steps:
Step 1: Find the x-intercepts (where y=0).
Set the quadratic equation equal to zero and solve for x:
2x^2 -15x+18 = 0
You can factor the equation or use the quadratic formula to find the x-intercepts:
(x-2)(2x-9) = 0
Setting each factor equal to zero, we get:
x - 2 = 0 or 2x - 9 = 0
Solving for x, we get:
x = 2 or x = 4.5
So, the x-intercepts are x = 2 and x = 4.5.
Step 2: Determine the intervals between the x-intercepts.
To find the intervals on the x-axis, we need to determine the sign of the quadratic function between the x-intercepts.
Let's choose a test point in each interval:
For the interval (-∞, 2), we can plug in x = 0.
For the interval (2, 4.5), we can plug in x = 3.
For the interval (4.5, +∞), we can plug in x = 5.
Step 3: Evaluate the function at the test points.
We substitute the test points into the quadratic equation and determine the sign of y.
For x = 0:
y = 2(0)^2 - 15(0) + 18 = 18
Since y > 0, the quadratic is positive in the interval (-∞, 2).
For x = 3:
y = 2(3)^2 - 15(3) + 18 = 9
Since y > 0, the quadratic is positive in the interval (2, 4.5).
For x = 5:
y = 2(5)^2 - 15(5) + 18 = -17
Since y < 0, the quadratic is not positive in the interval (4.5, +∞).
Based on these evaluations, the quadratic function y=2x^2 -15x+18 is positive in the intervals (-∞, 2) and (2, 4.5).
To determine the intervals on which a quadratic function is positive, we need to find the values of x that make the function greater than zero. In other words, we need to find the x-values for which the function is above the x-axis.
Given the quadratic function y = 2x^2 - 15x + 18, we can solve for x by setting the function equal to zero and then factoring or using the quadratic formula. However, in this case, we can use a different method, called factoring by grouping, to determine the intervals when the function is positive without actually finding the exact values of x.
Step 1: Factor the quadratic expression.
We have y = 2x^2 - 15x + 18. To factor this expression, we look for two numbers that multiply to give us 2 * 18 = 36 and add up to -15 (the coefficient of the middle term). These numbers are -3 and -12.
Using these numbers, we rewrite the quadratic expression as follows:
y = 2x^2 - 3x - 12x + 18
Step 2: Group the terms and factor by grouping.
Now, we group the terms as follows:
y = (2x^2 - 3x) + (-12x + 18)
Step 3: Factor out common factors from each group.
In the first group, we can factor out an x:
y = x(2x - 3) + (-12x + 18)
In the second group, we can factor out -6 (to retain the negative sign for consistency):
y = x(2x - 3) - 6(2x - 3)
Step 4: Factor out the common binomial.
Now, we can factor out (2x - 3) as it appears in both terms:
y = (2x - 3)(x - 6)
Step 5: Set each factor equal to zero and solve for x.
To determine the intervals on which the function is positive, we need to find the x-values that make the function greater than zero. For this quadratic function, we can set each factor equal to zero: (2x - 3) = 0 and (x - 6) = 0.
Solving the first equation, we find:
2x - 3 = 0
Adding 3 to both sides:
2x = 3
Dividing both sides by 2:
x = 3/2 or x = 1.5
Solving the second equation, we find:
x - 6 = 0
Adding 6 to both sides:
x = 6
Step 6: Determine the intervals.
Now that we have the x-values where the function is equal to zero, we can determine the intervals on which the quadratic function is positive.
We know that the function will be positive between the roots of the quadratic function. In this case, the roots are x = 1.5 and x = 6. Therefore, the function will be positive in the intervals (1.5, 6).
So, the intervals on which the quadratic function y = 2x^2 - 15x + 18 is positive are (1.5, 6).