Solving Quadratics

By Arianne Goode

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You can solve quadratics in 2 different ways:

1. by factoring

Arianne Goode Factoring

...or...

2. by using the quadratic formula.

Arianne Goode Quadratic Formula

You can also graph quadratics...

in standard form ↓...

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...or in vertex form ↓.

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There are also multiple ways to solve complex numbers:

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Dividing:

  1. Multiply by conjugate, switching the addition/subtraction sign on the bottom, if needed
  2. Use foil method to multiply
  3. Remember that i² equals -1
  4. Combine like terms
  5. Reduce if possible

Last comes solving application problems. There are two main types of these:

1. "Maximum/Minimum" problems

Example: A paper plane is thrown out of a window. The height in feet, h, of the plane above the ground is given by h = -5t² + 30t + 100: t is the time in seconds. Find the maximum height of the paper plane in the air.


  1. You must plug the numbers into the vertex formula to find what time the paper plane will reach its maximum height. Make sure to reduce if possible.
  2. Now you have the time that the plane will reach its maximum height, but you need to know the actual maximum height. So, you need to plug the vertex back into the original equation, having it take the place of t.
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2. "Zeros (time that an object hits the ground)" problems

Example: The height in feet, h, of a kite above the ground is given by h = -10t² + 70t + 200: t is the time in seconds. Find the time it takes the kite to strike the ground.


  1. The question asks how long it will take for the kite to fall to ground level, or for the height to equal 0, so you need to set h = 0 and solve for t.
  2. Plug the numbers into the quadratic formula and solve, making sure to reduce if possible.
  3. You will end up with two answers: one positive and one negative. The negative answer won't work, however, because you cannot have a negative amount of time. So, the positive answer is the right one.
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