# The Quadratic Equation

**Need To Know
**▪ Review the quadratic equation

▪ The Principle of Square Roots

▪ Completing the Square

**The Principle of Square Roots**

The Principle of Square Roots

For all positive real numbers b ,

Reminder: ±

If X^{2} = k, then

In Words

We use the square root to remove “squared stuff”.

Always remember there are two answers ,

so use the +.

Solve:

9x^{2} + 16 = 0

(3a – 12)^{2} = 18

**Completing the Square**

Figure out what constant term to add to make the

polynomial factor into a perfect square.

x^{2} + 4x + ____ = ( )^{2}

x^{2} – 10x + ____ = ( )^{2}

x^{2} – 24x + ____ = ( )^{2}

x^{2} + 3x + ____ = ( )^{2}

x^{2} + bx + ____ = ( )^{2}

Solve by completing the square:

3y^{2} + 12y + 6 = 0

How to Solve by

Completing the Square

1 If **a** is not 1 divide

by **a** on both sides.

2. Put equation in

x ^{2} + bx = c form.

3. Add the complete

square number to

both sides.

4. Solve the equation

by the square root

method

Solve by completing the square:

2x^{2} – 2x + 1 = 0

How to Solve by

Completing the Square

1 If a is not 1 divide

by a on both sides.

2. Put equation in

ax^{2} + bx = c form.

3. Add the complete

square number to

both sides.

4. Solve the equation

by the square root

method

**8.1 Conclusion**

**Ways to Solve Quadratic Equations
**1. Factoring method

(Set up: equation must equal zero )

2. Square root method

(Set up: “squared stuff” by itself)

3. Completing the square method

(Set up: the leading coefficient = 1)

Rating | Doable |

Easy | Not always |

Easy | Not always |

end

**8.2 The Quadratic Formula **

**Need To Know
**▪ The parts of the quadratic equation

▪ The Quadratic Formula and how to use it

**The Quadratic Equation**

**Example: Find the coefficients.**

3x^{2} - 7x + 11 = 0

**For ax ^{2} + bx + c = 0,**

Solve:

y^{2} + 13 = 6y

How To Solve

1. Put equation in

standard form

(it must equal 0).

2. Find a, b, c

3. Plug into the

formula & simplify

Solve:

x^{2} – 4x + 4 = 5

How To Solve

1. Put equation in

standard form

(it must equal 0).

2. Find a, b, c

3. Plug into the

formula & simplify

Solve:

r^{2} = -3r + 8

How To Solve

1. Put equation in

standard form

(it must equal 0).

2. Find a, b, c

3. Plug into the

formula & simplify

**8.2 Conclusion**

**Ways to Solve Quadratic Equations
**1. Factoring method

(Set up: equation must = 0)

2. Square root method

(Set up: “squared stuff” by itself)

3. Completing the square method

(Set up: the leading coefficient = 1)

4. Quadratic Formula

(Set up: equation must = 0)

Rating | Doable |

Easy | Not always |

Easy | Not always |

Hard | Always |

end

**8.3 Applications with Quadratics**

**Need To Know
**▪ Review methods of solving quadratics

▪ Applications

▪ Solving quadratic formulas

**8.3 Methods to Solve Quadratics**

**Ways to Solve Quadratic Equations
**1. Factoring method

(Set up: equation must = 0)

2. Square root method

(Set up: “squared stuff” by itself)

3. Completing the square method

(Set up: the leading coefficient = 1)

4. Quadratic Formula

(Set up: equation must = 0)

Rating | Doable |

Easy | Not always |

Easy | Not always |

Hardest | Always |

Hard | Always |

**Word Problems**

Peter’s car travels 200 miles averaging a certain speed.

If the car had gone 10 mph faster, the trip would have

taken 1 hour less. Find Peter’s average speed

During the first part of the trip, Mita’s car traveled 120

miles at a certain speed. Mita then drove another 100

miles at a speed that was 10 mph slower.

Her total trip took 4 hours. What were her speeds?

end

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