Solving Quadratic Equations

Terminology

1. A Quadratic equations is an equation that contains a second-degree term and no term of a higher degree.

2. The standard form of a quadratic equation is ,quad1.gif where a, b & c are real numbers and .

 

Steps for Solving Quadratic Equations by Factoring

1. Write the equation in standard form:

2. Factor completely.

3. Apply the Zero Product Rule , by setting each factor containing a variable to zero.  If ab = 0, then a = 0 or b = 0.                                     

4. Solve the linear equations in step 3.

5. Check.

Note: Most quadratic equations have 2 solutions . The 2 solutions correspond to  the x-intercepts of the graph of a quadratic function.

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Watch the Video: Solving Quadratic Equations by Factoring Basic Examples by PatrickJMT

 

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Watch the Video: Solving Quadratic Equations by Factoring another Example by PatrickJMT

 

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Watch the Video: Math Help Quadratics: Solve by Factoring by Pat McKeague

 

 

Check Yourself: Click on Activity

 Toggle open/close quiz question

Match the items.
The task is to match the lettered items with the correct numbered items. Appearing below is a list of lettered items. Following that is a list of numbered items. Each numbered item is followed by a drop-down. Select the letter in the drop down that best matches the numbered item with the lettered alternatives.
a. 
b. 7 & -4
c. -4 & -6
d. 
 

 

Steps for solving Quadratic application problems:

1. Draw and label a picture if necessary.

2. Define all of the variables.

3. Determine if there is a special formula needed. Substitute the given information into the equation.

4. Write the equation in standard form.

5. Factor.

6. Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers. (Hint: We can't have -5 ft. of carpet.)

7. Check your answers.

 

 Area of a rectangle and Landscaping/border/frame problems . 

Example 1:A vacant rectangular lot is being turned into a community vegetable garden measuring 8 meters by 12 meters. A path of uniform width is to surround garden. If the area of the lot is 140 square meters, find the width of the path surrounding the garden.

Step 1:Draw and label a picture if necessary.

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Step 2:Define all of the variables.

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Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

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Step 4:Write the equation in standard form.

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Step 5:Factor.

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Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

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Step 7:Check your answers.

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Example 2:Each side of a square is lengthened by 7 inches. The area of this new larger square is 81 square inches. Find the length of a side of the original square.

Step 1:Draw and label a picture if necessary.

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Step 2:Define all of the variables.

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Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

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Step 4:Write the equation in standard form.

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Step 5:Factor.

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Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

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Step 7:Check your answers.

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Pythagorean Theorem Problems: 

Example 3:A guy wire is attached to a tree to help it grow straight. The length of the wire is 2 feet greater than the distance from the base of the tree to the stake. The height of the wooden part of the tree is 1 foot greater than the distance from the base of the tree to the stake.

Step 1:Draw and label a picture if necessary.

 

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Step 2:Define all of the variables.

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Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

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Step 4:Write the equation in standard form.

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Step 5:Factor.

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Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

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Step 7:Check your answers.

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Example 5:A piece of wire measuring 20 feet is attached to a telephone pole as a guy wire. The distance along the ground from the bottom of the pole to the end of the wire is 4 feet greater than the height where the wire is attached to the pole. How far up the pole does the guy wire reach?

Step 1:Draw and label a picture if necessary.

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Step 2:Define all of the variables.

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Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

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Step 4:Write the equation in standard form.

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Step 5:Factor.

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Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

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Step 7:Check your answers.

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Motion Problems using the formula

Example 4:You throw a ball straight up from a rooftop 384 feet high with an initial speed of 3 feet per second. The function

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describes the height of the ball above the ground, s (t), in feet, t seconds after you threw it. The ball misses the rooftop on its way down and eventually strikes the ground. How long will it take for the ball to hit the ground?

Step 1:Draw and label a picture if necessary.

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Step 2:Define all of the variables.

 t = time, s(t) = height

Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

The formula that was given. 

Step 4:Write the equation in standard form.

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Step 5:Factor.

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Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

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Step 7:Check your answers.

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Example 5:Use the same function

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to determine when the height of the ball is 336 feet. 

Step 1:Draw and label a picture if necessary.

 factor5.jpg

Step 2:Define all of the variables.

 t = time, s(t) = height

Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

 The formula that was given

Step 4:Write the equation in standard form.

106.png  

Step 5:Factor.

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Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

 107.png

 

Step 7:Check your answers.

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Solve a Quadratic Equation by COMPLETING THE SQUARE .

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Watch The Video: Solving Quadratic Equations by Completing the Square by Patrick JMT

 

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Watch the Video: Quadratics: Completing the Square by Pat McKeague

 

Solving Quadratic Equations using the Quadratic Formula

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Example:

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Watch the Video: Solving Quadratic Equations using the Quadratic Formula-Example 3 by Patrick JMT

 

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Watch the Video: Math Help Quadratics: The Quadratic Formula by Pat McKeague

 

 

Graphing QuadraticFunctions

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Watch the Video: Graphing Quadratic Functions by Patrick JMT

 

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Click on this applet: Quadratic Function Calculator

 

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Click on the this applet: Quadratic Functions(General form)

 

 

Graphing Quadratic Inequalities

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