Write x² - 8x + 21 in the form (x + a)² + b where a and b are integers.
Understand the Problem
The question is asking us to rewrite the quadratic expression x² - 8x + 21 in the form (x + a)² + b, where a and b are integers. This involves completing the square.
Answer
The expression can be rewritten as \( (x - 4)^2 + 5 \).
Answer for screen readers
The expression ( x^2 - 8x + 21 ) can be rewritten as ( (x - 4)^2 + 5 ).
Steps to Solve
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Identify coefficients We start with the quadratic expression ( x^2 - 8x + 21 ). The coefficient of ( x ) is (-8).
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Complete the square Take half of the coefficient of ( x ) (which is (-8)), square it, and add/subtract in the expression. Half of (-8) is (-4), and squaring it gives: $$ (-4)^2 = 16 $$
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Rewrite the expression We can rewrite the quadratic expression by adding and subtracting (16): $$ x^2 - 8x + 16 - 16 + 21 $$
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Simplify the expression Now simplify the expression by grouping: $$ (x^2 - 8x + 16) + 5 $$ This can be written as: $$ (x - 4)^2 + 5 $$
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Identify values of (a) and (b) Now we see that we have expressed the quadratic in the form ( (x + a)^2 + b ): Here, ( a = -4 ) and ( b = 5 ).
The expression ( x^2 - 8x + 21 ) can be rewritten as ( (x - 4)^2 + 5 ).
More Information
Completing the square is a useful technique in algebra to convert quadratic expressions into vertex form. This form makes it easier to graph a parabola and understand its properties.
Tips
- Forgetting to adjust the constant after adding the squared term. Always ensure to balance the equation by subtracting the same value you added.
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