Completing the Square Practice
Rewrite ax² + bx + c as a(x − h)² + k: factor a out of the x terms, then add and subtract the square of half the new x coefficient.
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Write in vertex form.
12 problems
- 1.3x² + 6x + 7Answer3×(x+1)²+4
Take 3 out of the x terms
3×(x²+2×x)+7Complete the square
3×((x+1)²−1)+7Expand
3×(x+1)²−3+7Work out 3×(x+1)²−3+7
3×(x+1)²+4
- 2.−x² + 2x − 16Answer−(x−1)²−15
Take −1 out of the x terms
−(x²−2×x)−16Complete the square
−((x−1)²−1)−16Expand
−(x−1)²+1−16Work out −(x−1)²+1−16
−(x−1)²−15
- 3.5x² − 20x + 13Answer5×(x−2)²−7
Take 5 out of the x terms
5×(x²−4×x)+13Complete the square
5×((x−2)²−4)+13Expand
5×(x−2)²−20+13Work out 5×(x−2)²−20+13
5×(x−2)²−7
- 4.−3x² + 6x − 5Answer−3×(x−1)²−2
Take −3 out of the x terms
−3×(x²−2×x)−5Complete the square
−3×((x−1)²−1)−5Expand
−3×(x−1)²+3−5Work out −3×(x−1)²+3−5
−3×(x−1)²−2
- 5.−x² + 8x − 9Answer−(x−4)²+7
Take −1 out of the x terms
−(x²−8×x)−9Complete the square
−((x−4)²−16)−9Expand
−(x−4)²+16−9Work out −(x−4)²+16−9
−(x−4)²+7
- 6.7x² − 14x + 13Answer7×(x−1)²+6
Take 7 out of the x terms
7×(x²−2×x)+13Complete the square
7×((x−1)²−1)+13Expand
7×(x−1)²−7+13Work out 7×(x−1)²−7+13
7×(x−1)²+6
- 7.−3x² + 6x − 7Answer−3×(x−1)²−4
Take −3 out of the x terms
−3×(x²−2×x)−7Complete the square
−3×((x−1)²−1)−7Expand
−3×(x−1)²+3−7Work out −3×(x−1)²+3−7
−3×(x−1)²−4
- 8.−x² + 2x − 9Answer−(x−1)²−8
Take −1 out of the x terms
−(x²−2×x)−9Complete the square
−((x−1)²−1)−9Expand
−(x−1)²+1−9Work out −(x−1)²+1−9
−(x−1)²−8
- 9.5x² − 20x + 8Answer5×(x−2)²−12
Take 5 out of the x terms
5×(x²−4×x)+8Complete the square
5×((x−2)²−4)+8Expand
5×(x−2)²−20+8Work out 5×(x−2)²−20+8
5×(x−2)²−12
- 10.−3x² + 6x − 6Answer−3×(x−1)²−3
Take −3 out of the x terms
−3×(x²−2×x)−6Complete the square
−3×((x−1)²−1)−6Expand
−3×(x−1)²+3−6Work out −3×(x−1)²+3−6
−3×(x−1)²−3
- 11.5x² − 10x + 12Answer5×(x−1)²+7
Take 5 out of the x terms
5×(x²−2×x)+12Complete the square
5×((x−1)²−1)+12Expand
5×(x−1)²−5+12Work out 5×(x−1)²−5+12
5×(x−1)²+7
- 12.−3x² + 6x − 8Answer−3×(x−1)²−5
Take −3 out of the x terms
−3×(x²−2×x)−8Complete the square
−3×((x−1)²−1)−8Expand
−3×(x−1)²+3−8Work out −3×(x−1)²+3−8
−3×(x−1)²−5
The vertex form shows the parabola's turning point, (h, k), at a glance.
Common questions
- What is completing the square?
- Rewriting a quadratic so it contains a perfect square: a(x − h)² + k.
- What is it used for?
- Finding the vertex of a parabola, and deriving the quadratic formula.
