Determine two of the values of b so that expression x^2 +bx -12 can be factored
Just work it out by using diff. factors of 12. Also one binomial will be subtraction and the other addition.
x^2 +bx -12
The discriminant is b^2+48
So, you need b^2+48 to be a perfect square.
1+48 = 49, so b = ±1 works
16+48=64, so b = ±4 works
121+48 = 169, so b = ±11 works
To determine two values of "b" such that the expression x^2 + bx - 12 can be factored, we can use the fact that for a quadratic expression to be factored, its discriminant must be a perfect square (non-negative).
The discriminant of the quadratic expression ax^2 + bx + c is given by b^2 - 4ac.
In this case, the quadratic expression is x^2 + bx - 12, so the discriminant is b^2 - 4(1)(-12) = b^2 + 48.
To find two values of "b" such that the discriminant is a perfect square, we can set the discriminant equal to a perfect square and solve for "b".
b^2 + 48 = k^2
Where "k" represents a perfect square.
Rearranging the equation, we have:
b^2 - k^2 = -48
Applying the difference of squares factorization, we get:
(b + k)(b - k) = -48
Now, we can try different values of "k" that would give us integer values for "b".
For example, if k = 1, then we have:
(b + 1)(b - 1) = -48
Simplifying further, we get:
b^2 - 1 = -48
Adding 1 to each side:
b^2 = -47
Since we cannot have a non-negative value for b^2 when dealing with real numbers, this value of "k" does not give us valid values for "b".
We can continue trying different values of "k" until we find two values of "b" that satisfy the equation.
Let's try k = 2:
(b + 2)(b - 2) = -48
Simplifying further:
b^2 - 4 = -48
Adding 4 to each side:
b^2 = -44
Again, we cannot have a non-negative value for b^2 when dealing with real numbers. So this value of "k" also does not give us valid values for "b".
We can continue this process by trying various values of "k" until we find two values of "b" that satisfy the equation.
To determine the values of b for which the expression x^2 + bx - 12 can be factored, we need to find two numbers that multiply to give -12 and add up to the coefficient of the middle term (bx).
Let's call these two numbers a and d.
We know that a * d = -12, so we need to find the factors of -12. The factors of -12 are:
-1, 12
-2, 6
-3, 4
1, -12
2, -6
3, -4
Next, we need to find the pair of factors whose sum is equal to b.
Considering the equation x^2 + bx - 12, the coefficient of x in the equation is b. Hence, the sum of the factors should be equal to b.
So, to find the values of b, we sum up the factors of -12 and check if any of those sums match b.
For the pairs of factors:
-1 + 12 = 11
-2 + 6 = 4
-3 + 4 = 1
1 + (-12) = -11
2 + (-6) = -4
3 + (-4) = -1
Since none of these sums match b, there are no two values of b for which the expression x^2 + bx - 12 can be factored.