Write x^2 + 8x + 19 in the form (x + a)^2 + b where a and b are integers.
Understand the Problem
The question is asking to rewrite the quadratic expression x^2 + 8x + 19 in the completed square form (x + a)^2 + b, identifying the integer values for a and b.
Answer
The completed square form is \( (x + 4)^2 + 3 \).
Answer for screen readers
The completed square form is ( (x + 4)^2 + 3 ) where ( a = 4 ) and ( b = 3 ).
Steps to Solve
- Identify the quadratic expression
We start with the expression ( x^2 + 8x + 19 ).
- Complete the square for ( x^2 + 8x )
To complete the square, take the coefficient of ( x ), which is 8, divide it by 2 to get 4, and then square it:
[ 4^2 = 16 ]
Thus, we rewrite ( x^2 + 8x ) as:
[ (x + 4)^2 - 16 ]
- Substitute back into the expression
Now we substitute ( (x + 4)^2 - 16 ) back into the original expression:
[ (x + 4)^2 - 16 + 19 ]
- Simplify the expression
Now simplify:
[ (x + 4)^2 + 3 ]
This shows that ( a = 4 ) and ( b = 3 ).
The completed square form is ( (x + 4)^2 + 3 ) where ( a = 4 ) and ( b = 3 ).
More Information
Completing the square is a method used to rewrite quadratic expressions in a form that makes it easier to analyze properties like the vertex of the parabola.
Tips
- Forgetting to adjust the constant: When completing the square, it's crucial to adjust the constant term accordingly. Ensure that you add and subtract the square of half the coefficient of ( x ).
- Not simplifying: Sometimes, students forget to combine like terms after completing the square.
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