The sum of the roots of a
quadratic equation is 5 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
To find the quadratic equation, we can use the fact that the sum of the roots is equal to the negation of the coefficient of the linear term and the product of the roots is equal to the constant term.
Let's call the roots of the quadratic equation a and b.
According to the given information:
a + b = 5 ------(1)
ab = 4 ------(2)
We can solve these equations to find the values of a and b.
From equation (1), we can express a in terms of b:
a = 5 - b
Substituting this into equation (2), we get:
(5 - b)b = 4
5b - b^2 = 4
b^2 - 5b + 4 = 0
Now, we have a quadratic equation in terms of b. We can factorize it to find the values of b.
(b - 1)(b - 4) = 0
So, b = 1 or b = 4
If b = 1, substitute this value back into equation (1) to find a:
a + 1 = 5
a = 4
If b = 4, substitute this value back into equation (1) to find a:
a + 4 = 5
a = 1
Therefore, the roots of the quadratic equation are 4 and 1.
The quadratic equation with roots 4 and 1 can be expressed as:
(x - 4)(x - 1) = 0
x^2 - 5x + 4 = 0
Therefore, the quadratic equation is option C: 2x^2 - 8x + 5 = 0.
To find the quadratic equation with the given sum and product of roots, we can use the formula for the sum and product of roots of a quadratic equation.
Let's denote the roots of the quadratic equation as x1 and x2.
1. Sum of the roots:
The sum of the roots of a quadratic equation is given by the formula:
Sum of roots = (-b) / a
where a and b are the coefficients of the quadratic equation.
From the given information, we have the sum of roots as 5. Therefore, we can write:
(-b) / a = 5
2. Product of the roots:
The product of the roots of a quadratic equation is given by the formula:
Product of roots = c / a
where c is the constant term in the quadratic equation.
From the given information, we have the product of roots as 4. Therefore, we can write:
c / a = 4
Now, let's analyze the answer choices:
A. 2x^2 + 5x + 8 = 0
Here, the coefficient of x^2 is 2, the coefficient of x is 5, and the constant term is 8.
Using the formulas above, we can calculate:
(-5) / 2 = -2.5 (not equal to 5)
Therefore, option A is not the correct answer.
B. 2x^2 - 5x + 8 = 0
Here, the coefficient of x^2 is 2, the coefficient of x is -5, and the constant term is 8.
Using the formulas above, we can calculate:
(-(-5)) / 2 = 5/2 = 2.5 (not equal to 5)
Therefore, option B is not the correct answer.
C. 2x^2 - 8x + 5 = 0
Here, the coefficient of x^2 is 2, the coefficient of x is -8, and the constant term is 5.
Using the formulas above, we can calculate:
(-(-8)) / 2 = 8/2 = 4 (not equal to 5)
Therefore, option C is not the correct answer.
D. 2x^2 + 8x - 5 = 0
Here, the coefficient of x^2 is 2, the coefficient of x is 8, and the constant term is -5.
Using the formulas above, we can calculate:
(-8) / 2 = -4
and
(-5) / 2 = -2.5
The given sum of the roots is 5, so the sum of the roots of this quadratic equation is not equal to 5.
Therefore, option D is not the correct answer.
Based on our analysis, none of the given options satisfy the conditions of the sum and product of roots given in the question. Please check the question or the given options for accuracy.