Question

Both roots of the quadratic equation x²-49x c=0 are prime numbers. Find the value of c.

Answer 1

The provided quadratic equation does not have two distinct prime roots that multiply to give c; if we allow a repeated root, c would be 49.

Step-by-step explanation:

We are given a quadratic equation of the form x² - 49x + c = 0 where the roots are prime numbers. To find the value of c, we can utilize the fact that in a quadratic equation ax² + bx + c = 0, the product of the roots is equal to c/a. Here, a is 1, so the product of the roots is equal to c. Since both roots are prime numbers, and one of their products gives us c, we look at the coefficient of x, which is -49. Factoring -49, we get -7 x 7, which suggests the roots are 7 and 7 because the sum of the roots (which is -b/a), must equal 49. But since a prime number can't be a duplicate root in a quadratic equation, this problem has no solutions where both roots are distinct prime numbers, as the prime numbers can't add up to 49 without duplicating one of them. Hence, either the question is flawed, or we are looking for duplicated prime roots, which means c would simply be 7 x 7 = 49.

Answer 2

To find the value of c, we can use the fact that both roots of the quadratic equation x²-49x+c=0 are prime numbers. By using the quadratic formula and knowing that the discriminant must be a perfect square, we can solve for c. The value of c that satisfies the conditions is 1.

Step-by-step explanation:

To find the value of c, we can use the fact that both roots of the quadratic equation x² - 49x + c = 0 are prime numbers. Let's use the quadratic formula to solve for x.

The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = -49, and c is the value we want to find. Since the roots are prime numbers, we know that the discriminant (b² - 4ac) must be a perfect square.

Let's substitute the values into the formula and solve for c.

x = (-(-49) ± √((-49)² - 4(1)(c))) / (2(1))

x = (49 ± √(2401 - 4c)) / 2

The value of c that makes the discriminant a perfect square is 1.