1. Given the following polynomial: 2x^2 + 7x - 15 = 0 Check all that apply.
The value of the discriminant is 169.
There are 2 real roots.
There are 2 irrational roots.
The graph intersects the y-axis twice.
The parabola is directed upward.
The axis of symmetry is located at: x = -7/4
The vertex is located at: (-7/4, -49/8)
The roots are: {5,3/2}
The graph intersects the y axis at (0, -15).
The graph intersects the x-axis at (-5, 0) and (1.5, 0)
II.
Use the quadratic formula to solve the following: 3x^2 - x + 2 = 0
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2x^2 + 7x - 15 = (2x-3)(x+5)
To determine which statements are true for the given polynomial 2x^2 + 7x - 15 = 0, we can analyze the equation and use various properties and formulas. Let's go through each statement one-by-one:
1. The value of the discriminant is 169.
The discriminant of a quadratic equation in the form ax^2 + bx + c = 0 is given by b^2 - 4ac. In this case, a = 2, b = 7, and c = -15. Therefore, the discriminant is calculated as (7^2) - 4(2)(-15) = 49 + 120 = 169. This statement is true.
2. There are 2 real roots.
In order to determine the number of real roots, we can check the value of the discriminant. If the discriminant is positive, there are two distinct real roots. Since the value of the discriminant is 169, which is positive, this statement is true.
3. There are 2 irrational roots.
If the discriminant is not a perfect square, the roots will be irrational. Since the discriminant is 169, which is a perfect square (13^2), the roots will be rational. Therefore, this statement is false.
4. The graph intersects the y-axis twice.
To find the y-intercept, we set x to 0 in the equation and solve for y. Plugging in x = 0, we get y = 0 - 15 = -15. Therefore, the graph intersects the y-axis at (0, -15) once, so this statement is false.
5. The parabola is directed upward.
The leading coefficient of the quadratic term is 2, which is positive. A positive leading coefficient indicates that the parabola opens upward. Therefore, this statement is true.
6. The axis of symmetry is located at x = -7/4.
The axis of symmetry for a parabola in the form y = ax^2 + bx + c is given by x = -b/(2a). In this case, a = 2 and b = 7. Plugging in these values, we get x = -(7)/(2*2) = -7/4. Therefore, this statement is true.
7. The vertex is located at (-7/4, -49/8).
The vertex of a parabola can be found using the formula x = -b/(2a) and y = f(x), where f(x) corresponds to the value of y when x is substituted into the equation. We have already determined that the axis of symmetry is x = -7/4. Plugging this value into the equation, we can find the corresponding y-value. Substituting x = -7/4 into the equation 2x^2 + 7x - 15 = 0, we get y = 2(-7/4)^2 + 7(-7/4) - 15. Simplifying, y equals -49/8. Therefore, this statement is true.
8. The roots are: {5, 3/2}.
To find the roots of the quadratic equation, we can use the quadratic formula, which is x = (-b ± √(b^2 - 4ac))/(2a). For our equation 2x^2 + 7x - 15 = 0, the values of a, b, and c are 2, 7, and -15 respectively. Substituting these values into the quadratic formula, we get x = (-7 ± √(7^2 - 4(2)(-15)))/(2(2)). Simplifying further, x equals (5, -3/2). Therefore, this statement is false.
9. The graph intersects the y-axis at (0, -15).
We have already determined that the y-intercept is -15. Therefore, this statement is true.
10. The graph intersects the x-axis at (-5, 0) and (1.5, 0).
To find the x-intercepts of a quadratic equation, we set y equal to 0 and solve for x. In this case, we want to find the values of x when 2x^2 + 7x - 15 = 0. Factoring or using the quadratic formula, we find that the equation can be written as (2x - 3)(x + 5) = 0. Setting each factor equal to zero, we get x = 3/2 and x = -5. Therefore, this statement is true.
Now let's move on to the second part of your question:
To solve the quadratic equation 3x^2 - x + 2 = 0 using the quadratic formula, we need to determine the values of a, b, and c. Comparing this equation to the standard form ax^2 + bx + c = 0, we can conclude that a = 3, b = -1, and c = 2. Plugging these values into the quadratic formula x = (-b ± √(b^2 - 4ac))/(2a), we get:
x = (-(-1) ± √((-1)^2 - 4(3)(2)))/(2(3))
= (1 ± √(1 - 24))/6
= (1 ± √(-23))/6
Since the discriminant of -23 is negative, the roots will be complex numbers. Therefore, the solutions to the equation 3x^2 - x + 2 = 0 are (1 ± √(-23))/6.