To express the quadratic expression [tex]\( x^2 - 6x - 15 \)[/tex] in the form [tex]\( (x - a)^2 - b \)[/tex], we need to complete the square. Here is a detailed step-by-step solution:
1. Start with the given expression:
[tex]\[ x^2 - 6x - 15 \][/tex]
2. Isolate the quadratic and linear terms:
We will focus on [tex]\( x^2 - 6x \)[/tex]. To complete the square, we need to add and subtract the same value so that this part becomes a perfect square trinomial.
3. Find the value to complete the square:
Take the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-6\)[/tex]), halve it, and then square it:
[tex]\[ \left(\frac{-6}{2}\right)^2 = (-3)^2 = 9 \][/tex]
Now, add and subtract this value inside the expression:
[tex]\[ x^2 - 6x + 9 - 9 - 15 \][/tex]
4. Rewrite the quadratic as a perfect square trinomial and combine constants:
The expression [tex]\( x^2 - 6x + 9 \)[/tex] is a perfect square trinomial, which can be written as [tex]\( (x - 3)^2 \)[/tex]:
[tex]\[ (x - 3)^2 - 9 - 15 \][/tex]
5. Simplify the constants:
Combine [tex]\(-9\)[/tex] and [tex]\(-15\)[/tex]:
[tex]\[ (x - 3)^2 - 24 \][/tex]
6. Conclusion:
The expression [tex]\( x^2 - 6x - 15 \)[/tex] can be written in the form [tex]\( (x - a)^2 - b \)[/tex] where [tex]\( a = 3 \)[/tex] and [tex]\( b = 24 \)[/tex]. Therefore, we have:
[tex]\[ x^2 - 6x - 15 = (x - 3)^2 - 24 \][/tex]
Thus, the expression [tex]\( x^2 - 6x - 15 \)[/tex] in the form [tex]\( (x - a)^2 - b \)[/tex] is [tex]\( (x - 3)^2 - 24 \)[/tex].
