Determine whetere each of the following trinomials is a perfect square. If it is, factor the trinomial.
x^2 - 24x + 48 I cannot get the 24x I've tried (x-24)(x - 2) What am I doing wrong?
It is not a square. THe factors are 21.797 and 2.20, approximately.
To determine whether the trinomial x^2 - 24x + 48 is a perfect square, you can use a specific property of perfect square trinomials. In a perfect square trinomial, the first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms.
Let's apply this property to the given trinomial x^2 - 24x + 48:
First, check if the first and last terms are perfect squares:
The first term, x^2, is a perfect square (x*x).
The last term, 48, is not a perfect square.
Since the last term is not a perfect square, the trinomial is not a perfect square.
Thus, your attempts to factor (x-24)(x-2) are incorrect.
To factor the trinomial x^2 - 24x + 48, you can use other factoring methods. You could try factoring out a common factor, or use the quadratic formula to find the roots of the trinomial.
To determine whether a trinomial is a perfect square, we can use the discriminant of the trinomial. The formula for the discriminant is: b^2 - 4ac, where a, b, and c are the coefficients of the trinomial in the form ax^2 + bx + c.
In this case, the trinomial is x^2 - 24x + 48. Let's check if it's a perfect square by calculating the discriminant.
a = 1 (the coefficient of x^2)
b = -24 (the coefficient of x)
c = 48
The discriminant is given by: (-24)^2 - 4(1)(48) = 576 - 192 = 384
Since the discriminant is positive (384 > 0), the trinomial is not a perfect square.
Now, to factor the trinomial x^2 - 24x + 48, we can use factoring by grouping or the quadratic formula. Let's use factoring by grouping in this case.
Step 1: Split the middle term (-24x) into two terms whose coefficients multiply to give the product of the first and last terms.
The first term is x^2, so the product of the first and last terms is x^2 * 48 = 48x^2. We need two terms that multiply to give 48x^2 and add up to -24x.
Looking at the factors of 48 (1, 2, 3, 4, 6, 8, 12, 16, 24, 48) and considering their combinations, we can see that there are no two terms whose coefficients multiply to give 48x^2 and add up to -24x.
Therefore, the trinomial x^2 - 24x + 48 cannot be factored further using simple integer coefficients.
The factors you attempted, (x-24)(x-2), are incorrect because they do not multiply back to the original trinomial, x^2 - 24x + 48.
Hence, the trinomial x^2 - 24x + 48 cannot be factored any further using simple integers.