Find 8 values of c that make the following expression factorable. x^2+3x+c. How did you know?
You need 2 numbers which have a sum or difference of 3, c will be their product
e.g.
1,2 , product = 2 , c = 2 ----- (x+1)(x+2) = x^2 + 3x + 2
5, -2, product = -10 , c = -10 ----- (x+5)(x-2) = x^2 + 3x - 10
7, -4 , product = -28 , c = -28 ----- (x+7)(x-4) = x^2 + 3x - 28
-1, 4 , ................ x^2 + 3x - 4
sofar I have
c = 2, -10, -28, -4, ....
continue
To find values of c that make the expression x^2+3x+c factorable, we need to consider the discriminant of the quadratic equation. The discriminant can be defined as Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2+bx+c.
In this case, a = 1, b = 3, and the quadratic equation is x^2+3x+c. For the expression to be factorable, the discriminant should be a perfect square (a number that can be expressed as the square of an integer).
For the equation x^2+3x+c, the discriminant is given by Δ = (3)^2 - 4(1)(c) = 9 - 4c.
Now, we want Δ to be a perfect square, so we need to find values of c that make 9 - 4c a perfect square.
Let's consider some perfect squares:
0^2 = 0
1^2 = 1
2^2 = 4
3^2 = 9
So the possible values for 9 - 4c are 0, 1, 4, 9. We can solve the equation 9 - 4c = 0, 9 - 4c = 1, 9 - 4c = 4, and 9 - 4c = 9 to find the corresponding values of c.
1. When 9 - 4c = 0:
Solving this equation, we find c = 9/4.
2. When 9 - 4c = 1:
Solving this equation, we find c = 2.
3. When 9 - 4c = 4:
Solving this equation, we find c = 5/4.
4. When 9 - 4c = 9:
Solving this equation, we find c = 0.
So, the values of c that make the expression x^2 + 3x + c factorable are c = 9/4, 2, 5/4, and 0.
We arrived at these values by examining the discriminant of the quadratic equation and identifying the possible values of c that would make it a perfect square. By solving the resulting equations, we found the corresponding values of c.