1
Solve by Completing the Square: x² − 4x − 7 = 0
1
Original Equation
x² − 4x − 7 = 0
We need to rearrange this equation into the form (x − h)² = k by completing the square.
2
Complete the Square
−(−2)² − 7 + (x − 2)² = 0
Take half of −4, which is −2, then square it: (−2)² = 4
How to complete the square:
1. Take the coefficient of x (which is −4)
2. Divide by 2: −4 ÷ 2 = −2
3. Square it: (−2)² = 4
4. Add and subtract this value to maintain balance
3
Simplify & Balance
−4 − 7 + (x − 2)² = 0
−11 + (x − 2)² = 0
(x − 2)² = 11
Add 11 to both sides
Balance method: Whatever you do to one side, do to the other.
We add 11 to both sides to isolate (x − 2)²
4
Take Square Root
x − 2 = ±√11
Remember: both positive and negative roots!
Important: When taking the square root of both sides, we must consider both the positive and negative square root.
This gives us two equations:
x − 2 = √11 OR x − 2 = −√11
5
Solve for x
x = 2 + √11 OR x = 2 − √11
Add 2 to both sides of each equation
Final step: Add 2 to both sides to isolate x.
x = √11 + 2 OR x = −√11 + 2
6
Calculate (3 s.f.)
x = 5.32 OR x = −1.32
√11 ≈ 3.3166...
x = 2 + 3.3166... ≈ 5.32 (3 s.f.)
x = 2 − 3.3166... ≈ −1.32 (3 s.f.)
y = x² − 4x − 7 =
−7.0
Check your reasoning
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0/4
currently correct
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2
Why does (x − 2)² = 11 give two equations?
3
What are the exact solutions?
4
What do the two solutions represent on the graph?
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