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Introduction
Introduction
On this page, we will learn how to do completing the square problems for only the cases of a = 1.
On this page, we will learn how to do completing the square problems for only the cases of a = 1.
Learning Objective: Express in Complete-the-Square Form
Learning Objective: Express in Complete-the-Square Form
Example: Express x² + 4x + 8 in the form (x + h)² + k, where h and k are constants.
Example: Express x² + 4x + 8 in the form (x + h)² + k, where h and k are constants.
Step 1: Identify 'b'
Step 1: Identify 'b'
Half the value of b = 4/2 = + 2.
Half the value of b = 4/2 = + 2.
This tells us that the big square will be (x + 2)² and the small square will be 2².
This tells us that the big square will be (x + 2)² and the small square will be 2².
Step 2: Big Square minus Small Square
Step 2: Big Square minus Small Square
Rewrite the expression by changing x² + 4x into the big square (x + 2)² minus the small square 2².
i.e. x² + 4x + 8 = (x + 2)² − 2² + 8
Step 3: Simplify
Step 3: Simplify
Simplify the expression x² + 4x + 8 = (x + 2)² + 4.
Step 4: Check back
Step 4: Check back
Test the expression using π .
Use a calculator to evaluate π² + 4π + 8 and (π + 2)² + 4.
Do they both give the same value?
Problem: Solving using Complete-the-Square
Problem: Solving using Complete-the-Square
Example: Solve x² + 4x + 8 = 10.
Example: Solve x² + 4x + 8 = 10.
Step 1 - 4: Use the steps above
Step 1 - 4: Use the steps above
We will get (x + 2)² + 4 = 10
We will get (x + 2)² + 4 = 10
Step 5: Make the big square the subject
Step 5: Make the big square the subject
We subtract both sides by 4 to get (x + 2)² = 6
We subtract both sides by 4 to get (x + 2)² = 6
Step 6: Make x the subject
Step 6: Make x the subject
Do NOT forget the ± when you square-root both sides!
Square root both sides to get x + 2 = ±√6
Subtract 2 from both sides to get x = −2 ±√6
If the question ask for exact answers, this is your final answer.
Step 7: Present Answer
Step 7: Present Answer
In the absence of specific instructions, truncate your answer to 5 significant figures: x = −4.4494 or 0.44948 (5s.f)
Leave the final answer in 3 significant figures: x = −4.45 or 0.449 (3s.f.)
Step 8: Check
Step 8: Check
Test the answer by substituting the values back into the original equation.
Use a calculator to evaluate (0.449)² + 4(0.449) + 8. Is it close to 10? What about −4.45?
Next: Let's explore the graphical significance of h and k.
Next: Let's explore the graphical significance of h and k.
But what if a ≠ 1?
But what if a ≠ 1?
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