Quadratic equation whose roots are 5 and -5 with coefficient 3
using the letter x to represent the variable
please show steps
What did you get ?
3(x-5) (x+5)= 3x^2-28x + 75
Your expansion is incorrect
3(x-5)(x+5)
= 3(x^2 - 25)
= 3x^2 - 75
Since you wanted a quadratic equation, it should be
3x^2 - 75 = 0
To find the quadratic equation with roots 5 and -5, we can use the fact that the roots of a quadratic equation with the form ax^2 + bx + c = 0 are given by the equations:
x = (-b ± √(b^2 - 4ac))/(2a)
Since the roots are 5 and -5, we can substitute these values into the equation:
5 = (-b ± √(b^2 - 4ac))/(2a)
-5 = (-b ± √(b^2 - 4ac))/(2a)
Now, let's solve for b, the coefficient of the linear term, a, the coefficient of the quadratic term, and c, the constant term.
Using the first equation, where x = 5:
5 = (-b ± √(b^2 - 4ac))/(2a)
Simplifying, we have:
5 = (-b ± √(b^2 - 4ac))/(2a)
10a = -2b ± √(b^2 - 4ac)
10a + 2b = ± √(b^2 - 4ac)
(10a + 2b)^2 = b^2 - 4ac
Expanding and simplifying:
100a^2 + 40ab + 4b^2 = b^2 - 4ac
100a^2 + 40ab = -4ac
Dividing both sides by 4 gives us:
25a^2 + 10ab = -ac
Dividing through by a and simplifying:
25a + 10b = -c
Using the second equation, where x = -5, we repeat the same steps:
-5 = (-b ± √(b^2 - 4ac))/(2a)
Simplifying:
-5 = (-b ± √(b^2 - 4ac))/(2a)
-10a = -2b ± √(b^2 - 4ac)
-10a + 2b = ± √(b^2 - 4ac)
(-10a + 2b)^2 = b^2 - 4ac
Expanding and simplifying:
100a^2 - 40ab + 4b^2 = b^2 - 4ac
100a^2 - 40ab = -4ac
Dividing both sides by 4:
25a^2 - 10ab = -ac
Dividing through by a:
25a - 10b = -c
Now, we have two equations:
25a + 10b = -c
25a - 10b = -c
We can add these equations together to eliminate the c terms:
(25a + 10b) + (25a - 10b) = -c + (-c)
50a = -2c
Dividing through by -2 gives us:
-25a = c
So, we have found that the constant term, c, is equal to -25a.
Now, let's substitute this back into one of the original equations, for example, 25a + 10b = -c:
25a + 10b = -(-25a)
25a + 10b = 25a
10b = 0
This means that b must be equal to 0.
Finally, we substitute a = 3 (as given in the question) into the equation -25a = c:
-25(3) = c
-75 = c
Therefore, the quadratic equation with roots 5 and -5 and coefficient 3 is:
3x^2 + 0x - 75 = 0