12. The sum of the roots of a quadratic
equation is 5
2
and the product of its roots is
4. The quadratic equation is____
A. 2x2 + 5x + 8 = 0
B. 2x2 — 5x + 8 = 0
C. 2x2 — 8x + 5 = 0
D. 2x2 + 8x — 5 = 0
Let's denote the roots of the quadratic equation as x1 and x2.
First, we know that the sum of the roots is given by the formula:
x1 + x2 = -b/a
where a and b are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
In our equation, since the sum of the roots is 5/2, we have:
x1 + x2 = 5/2
Next, we know that the product of the roots is given by the formula:
x1 * x2 = c/a
In our equation, since the product of the roots is 4, we have:
x1 * x2 = 4
Now, let's find the equations based on the given information.
From the equation x1 + x2 = 5/2, we can rewrite it as:
x1 = 5/2 - x2
Plugging this into our product of the roots equation, we have:
(5/2 - x2) * x2 = 4
Simplifying, we get:
5x2 - 2x2^2 = 8
Rearranging the equation and simplifying further, we have:
2x2^2 - 5x2 + 8 = 0
Therefore, the quadratic equation is 2x^2 - 5x + 8 = 0, which corresponds to option B.
To find the quadratic equation with these given conditions, let's use some properties of quadratic equations:
1. The sum of the roots of a quadratic equation of the form ax^2 + bx + c = 0 is given by -b/a.
2. The product of the roots of a quadratic equation of the form ax^2 + bx + c = 0 is given by c/a.
Now let's apply these properties to the given conditions:
1. The sum of the roots is 5/2, which means -b/a = 5/2.
2. The product of the roots is 4, which means c/a = 4.
Let's write the equations based on these properties:
1. -b/a = 5/2
2. c/a = 4
Now let's examine the answer choices:
A. 2x^2 + 5x + 8
B. 2x^2 - 5x + 8
C. 2x^2 - 8x + 5
D. 2x^2 + 8x - 5
To match the given conditions, we need to find the equation that satisfies the equations we derived earlier.
Let's examine each option by comparing their coefficients with our derived equations:
A. Comparing with the equations, we have:
-b/a = -5/2 and c/a = 8/2 = 4/1
The coefficient of x^2 is 2, which is correct.
The coefficient of x is 5, which is not equal to -5/2.
B. Comparing with the equations, we have:
-b/a = 5/2 and c/a = 8/2 = 4/1
The coefficient of x^2 is 2, which is correct.
The coefficient of x is -5, which matches with -5/2.
C. Comparing with the equations, we have:
-b/a = 8/2 = 4 and c/a = 5/2
The coefficient of x^2 is 2, which is correct.
The coefficient of x is -8, which is not equal to 4.
D. Comparing with the equations, we have:
-b/a = 8/2 = 4 and c/a = -5/2
The coefficient of x^2 is 2, which is correct.
The coefficient of x is 8, which is not equal to 4.
By comparing the coefficients, we see that option B, 2x^2 - 5x + 8 = 0, matches with our derived equations.
Therefore, the quadratic equation is B. 2x^2 - 5x + 8 = 0.