 # 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 X2 = k, then In Words
We use the square root to remove “squared stuff”.
Always remember there are two answers ,
so use the +.

Solve:

9x2 + 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.

x2 + 4x + ____ = ( )2

x2 – 10x + ____ = ( )2

x2 – 24x + ____ = ( )2

x2 + 3x + ____ = ( )2

x2 + bx + ____ = ( )2

Solve by completing the square:

3y2 + 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:

2x2 – 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
ax2 + 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.

3x2 - 7x + 11 = 0

For ax2 + bx + c = 0, Solve:

y2 + 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:
x2 – 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:
r2 = -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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