Write the quadratic equation whose roots are 2 and −5, and whose leading coefficient is 3.
if R is a root, then (x-R) is a factor
3 * (x - 2) * (x + 5) = 3x^2 + 9x - 30
To write the quadratic equation with the given roots and leading coefficient, let's use the fact that for a quadratic equation in the form of ax^2 + bx + c = 0, the roots are given by the values of x where the equation equals zero.
Given that the roots are 2 and -5, we can set up two equations:
1) When x = 2, the equation equals zero:
3(2)^2 + b(2) + c = 0
2) When x = -5, the equation equals zero:
3(-5)^2 + b(-5) + c = 0
Simplifying these two equations:
1) 12 + 2b + c = 0
2b + c = -12 (Equation A)
2) 75 - 5b + c = 0
-5b + c = -75 (Equation B)
Now, we have a system of equations (Equation A and Equation B) with two variables (b and c). We can solve this system of equations to determine the values of b and c. Subtracting Equation B from Equation A, we can eliminate c:
(2b + c) - (-5b + c) = -12 - (-75)
2b + c + 5b - c = -12 + 75
7b = 63
b = 63 / 7
b = 9
Now, substitute the value of b back into Equation A to solve for c:
2(9) + c = -12
18 + c = -12
c = -12 - 18
c = -30
Therefore, the quadratic equation with roots 2 and -5, and a leading coefficient of 3 is:
3x^2 + 9x - 30 = 0