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Polynomials practice

Completing the square worksheets

Find the missing constant in ten monic quadratic trinomials. Choose whole, fractional or mixed linear coefficients and check exact reduced answers.

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Find the missing constant that makes each trinomial a perfect square.

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Turn a trinomial into a perfect square

Half the signed linear coefficient, then square that exact value. An odd whole coefficient can produce a fractional missing constant, and a fractional coefficient stays exact throughout the calculation.

Check the result by expanding the square: the middle coefficient must be twice the halved value, and the constant must be its square.

Learning objective

What this activity builds

Square half the linear coefficient to complete a monic quadratic as a perfect square.

Worked example

Complete x² − 11x + [blank].

121/4.

Half of −11 is −11/2. Squaring −11/2 gives 121/4, so the completed trinomial is (x − 11/2)².

Teaching notes

Make the practice count

  • Keep the sign when halving the linear coefficient.
  • Square both the numerator and denominator of a fraction.
  • Remember that squaring a negative number gives a positive constant.
  • Expand the resulting binomial square to check all three terms.

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Math SheetNameScore/ 10TeacherDateCompleting the SquareFind the missing constant that makes each trinomial a perfect square.1.y2-2y+2.y2-14y+3.a2+12a+4.c2-37c+5.p2+185p+6.x2+89x+7.s2+17s+8.a2+9a+9.s2-117s+10.y2-138y+math-sheet.comPage 1 / 1

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