# Completing the Square

To **complete the square** for the expression x^{2} + bx , add
, which is the square of

half the coefficient of x . Consequently,

When solving quadratic equations by completing the square,
you must add to both

sides to maintain equality.

**Completing the Square: Leading Coefficient is 1
**

Let’s solve the equation x

^{2}− 6x + 2 = 0 by completing the square.

x^{2} + 6x + 2 = 0 |
Original Equation |

x^{2} + 6x = −2 |
Subtract 2 from both sides |

Divide the 6 by 2, square it, and then add to both sides | |

x^{2} + 6x + 9 = 7 |
Simplify |

(x + 3)^{2} = 7 |
Perfect square trinomial |

Extract square roots | |

Solutions |

**Completing the Square: Leading Coefficient is Not 1
**

Let’s solve the equation 3x

^{2}− 4x − 5 = 0 by completing the square.

If the leading coefficient of a quadratic equation is not 1, you should divide both sides of

the equation by this coefficient before completing the square.

3x^{2} − 4x − 5 = 0 |
Original equation |

3x^{2} − 4x = 5 |
Add 5 to both sides |

Divide both sides by 3 | |

Divide by 2, square it, and then add to both sides | |

Perfect square trinomial | |

Extract square roots | |

Solutions |

Using a graphing calculator , you can see that the two
solutions are approximately

2.11963 and –0.78630, which agree with the two graphical solutions shown below.

**Completing the Square: One Term is Not Present
**

Let’s solve the equation 4x

^{2}− 7x = 0 by completing the square.

As you can see, we have no constant but we will treat the problem the same as if there

was a constant present . We skip the step of moving the constant over to the other side of

the equation and continue on from there.

4x^{2} − 7x = 0 |
Original equation |

Divide both sides by 4 | |

Divide by 2, square it, and then add to both sides | |

Perfect square trinomial | |

Extract square roots | |

Solutions |

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