quadratic equation who roots are
-3 and -4 whose leading coefficient is 2
use the letter x to represent the variable
that would simply be
f(x) = 2(x+3)(x+4)
= 2x^2 + 14x + 24
To find the quadratic equation with roots -3 and -4, and a leading coefficient of 2, we can use the fact that the roots of a quadratic equation can be determined from the factors of the equation.
The general form of a quadratic equation is:
ax^2 + bx + c = 0
Since the leading coefficient (a) is 2, we have:
2x^2 + bx + c = 0
We know that the roots are -3 and -4. For a quadratic equation, the roots correspond to the values of x when the equation equals zero. So, substituting the roots into the equation, we have:
When x = -3:
2(-3)^2 + b(-3) + c = 0
18 - 3b + c = 0 ---- (Equation 1)
When x = -4:
2(-4)^2 + b(-4) + c = 0
32 - 4b + c = 0 ---- (Equation 2)
Now, we have two equations (Equation 1 and Equation 2) with three variables (b and c). Therefore, we need another equation to solve for b and c.
To find the third equation, we can use the fact that the coefficient of x^2 is the leading coefficient. So, we have:
2x^2 + bx + c = 0
Comparing this with the general form of a quadratic equation, we can say that a = 2. Since the leading coefficient is 2, we can substitute this value into Equation 1:
2(-3)^2 + b(-3) + c = 2
18 - 3b + c = 2
-3b + c = -16 ---- (Equation 3)
Now, we have a system of three equations (Equation 1, Equation 2, and Equation 3) with three variables (b and c).
We can solve this system of equations to find the values of b and c. Substituting Equation 3 into Equation 2, we get:
32 - 4b + (-3b + c) = 0
-7b + c = -32 ---- (Equation 4)
Now, we have a system of two equations (Equation 1 and Equation 4) with two variables (b and c). We can solve this system to determine the values of b and c.
Subtracting Equation 3 from Equation 4, we get:
-7b + c - (-3b + c) = -32 - (-16)
-7b + 3b = -32 + 16
-4b = -16
b = 4
Substituting the value of b = 4 into Equation 3, we get:
-3(4) + c = -16
-12 + c = -16
c = -4
Finally, substituting the values of b = 4 and c = -4 into the general form of the quadratic equation, we get:
2x^2 + 4x - 4 = 0
Therefore, the quadratic equation with roots -3 and -4, and a leading coefficient of 2, is:
2x^2 + 4x - 4 = 0