if the Roots of a quadratic equation are 5 and -4 then form the quadratic equation.
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Answer:
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Answer:
The quadratic equation with roots 5 and -4 is:
x^2 - x + 20 = 0.
Step-by-step explanation:
If the roots of a quadratic equation are 5 and -4, we can form the equation using the fact that the roots of a quadratic equation are the values of x for which the equation equals zero.
Let's denote the quadratic equation as ax^2 + bx + c = 0.
Since the roots are 5 and -4, we can write two separate equations:
For x = 5: a(5)^2 + b(5) + c = 0
25a + 5b + c = 0
For x = -4: a(-4)^2 + b(-4) + c = 0
16a - 4b + c = 0
We now have a system of three equations:
25a + 5b + c = 0
16a - 4b + c = 0
To find the values of a, b, and c, we can substitute one of the equations into the other. Let's subtract the second equation from the first:
25a + 5b + c - (16a - 4b + c) = 0
25a - 16a + 5b + 4b = 0
9a + 9b = 0
9(a + b) = 0
From this equation, we can see that a + b = 0, which means a = -b.
Substituting -b for a in any of the previous equations, we can solve for the values of b and c.
Let's choose the first equation:
25a + 5b + c = 0
25(-b) + 5b + c = 0
-25b + 5b + c = 0
-20b + c = 0
We can choose any value for b, let's say b = 1. Then, we can solve for c:
-20(1) + c = 0
-20 + c = 0
c = 20
Now we have the values of a, b, and c:
a = -b = -1
b = 1
c = 20
Therefore, the quadratic equation with roots 5 and -4 is:
x^2 - x + 20 = 0.