Completing the square calculator
Complete the square: x² + 6x + 2
(x + 3)² − 7
The turning point is (−3, −7), a minimum.
- Roots
- x = −3 ± √7
Working
- Halve the coefficient of x: 6 ÷ 2 = 3, so write (x + 3)², which is x² + 6x + 9.
- Subtract the extra 9 and add back the constant: (x + 3)² − 7.
- Turning point: (−3, −7), a minimum because a is positive.
- Solving = 0: (x + 3)² = 7, so x + 3 = ± √7, giving x = −3 ± √7.
Your result
(x + 3)² − 7. The turning point is (−3, −7), a minimum.
Completing the square, step by step
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Your mock plan, in three taps
Completing the square on x² + 6x + 2 gives (x + 3)² − 7: halve the 6, then subtract 3², so the turning point is (−3, −7).
Questions people ask
How do you complete the square?
Halve the number in front of x and write (x + half)². That gives one extra (half)², so subtract it, then add the constant. x² + 8x + 5 = (x + 4)² − 16 + 5 = (x + 4)² − 11.
How do you complete the square when a is not 1?
Take a out of the x terms first. 2x² − 12x + 5 = 2(x² − 6x) + 5 = 2[(x − 3)² − 9] + 5 = 2(x − 3)² − 13.
How do you find the turning point from completed square form?
For (x + p)² + q the turning point is (−p, q): change the sign of the number in the bracket, keep the one outside. (x + 3)² − 7 turns at (−3, −7).
How do you solve a quadratic by completing the square?
Set the completed square equal to 0, move the constant over, square root both sides (remembering ±) and rearrange. (x + 3)² = 7, so x = −3 ± √7.
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How we work this out
What it does
Complete the square on any quadratic ax² + bx + c to find its turning point and exact roots, with every step shown.
Quadratics ax² + bx + c with whole-number a (not 0), b and c.
Method
- If a is not 1, factor it out of the x terms.
- Halve the x coefficient: x² + bx = (x + b/2)² − (b/2)².
- The turning point of a(x + p)² + q is (−p, q); setting it equal to 0 and square rooting gives the roots.