INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith

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• ## Related Books

 0 answersPost by John White on January 7 at 01:46:39 PMthe vertex is x = 1/2 not 1/4 2 answersLast reply by: Dr Carleen EatonTue May 3, 2016 7:11 PMPost by Kenosha Fox on May 2, 2016For example 1:solve be graphing,wasn't "X" 1/2 not 1/4 for the vertex ?? 2 answersLast reply by: Kenosha FoxSun May 1, 2016 3:03 PMPost by Kenosha Fox on April 29, 2016I have a question....in solving the One Double Root  example,what did "c"equal?

### Solving Equations by Graphing

• A quadratic equation is one that can be written in the form ax2 + bx + c = 0, where a ≠ 0. Its solutions are called the roots of the equation.
• The roots of a quadratic equation are x-intercepts of the graph of the related quadratic function.
• A quadratic equation has 2 real roots if its graph has two x-intercepts, 1 real root if it has one x-intercept (in this case, the graph is tangent to the x axis and the root is called a double root), and no real roots if it has no x-intercepts.
• If a root is not an integer, estimate the root by stating the two consecutive integers it lies between.
• In general, use graphing to solve an equation only if you would be satisfied with an estimate for the solutions, not exact values.
• A real number is a zero of the quadratic function f(x) if and only if it is a root of the equation f(x) = 0.

### Solving Equations by Graphing

Solve x2 + 4x = − 4 by graphing
• Find standard form ax2 + bc + cy = x2 + 4x + 4
• Find the vertex using x = − [b/2a]x = − ( [4/2(1)] ) = − 2y = ( − 2 )2 + 4( − 2 ) + 4 = 0
• Make a table of points
 x
 line
 − 2
 − 5
 − 4
 0
 1
 y
 line
 0
 9
 4
 4
 9
Graph

x = − 2
Solve − x2 = 6x by graphing
• Find standard form ax2 + bc + cy = − x2 − 6x
• Find the vertex using x = − [b/2a]x = − ( [( − 6)/(2( − 1))] ) = − 3y = − ( − 3 )2 − 6( − 3 ) = 9
• Make a table of points
 x
 line
 − 5
 − 4
 − 3
 − 2
 − 1
 y
 line
 5
 8
 9
 8
 5
Graph and find x intercepts

x = − 6,0
Solve 4x − x2 = − 5 by graphing
• Find standard form ax2 + bc + cy = − x2 + 4x + 5
• Find the vertex using x = − [b/2a]x = − ( [4/(2( − 1))] ) = 2y = − ( 2 )2 + 4( 2 ) + 5 = 9
• Make a table of points
 x
 line
 1
 2
 3
 4
 5
 y
 line
 8
 9
 8
 5
 0
Graph and find x - intercepts

x = − 1,5
Solve x([x/4] + 1) = 0 by graphing
• Find standard form ax2 + bx + cy = [(x2)/4] + x
• Find the vertex using x = − [b/2a]x = − ( [1/(2([1/4]))] ) = − 2y = [(( − 2 )2)/4] + ( − 2 ) = − 1
• Make a table of points
 x
 line
 − 4
 − 3
 − 2
 − 1
 0
 y
 line
 0
 − [3/4]
 − 1
 − [3/4]
 0

x = − 4,0
Solve [( − x2)/20] = − 6 by graphing
• Find standard form ax2 + bc + cy = [( − x2)/20] + 6
• Find the vertex using x = − [b/2a]x = − ( [0/(2( − 1))] ) = 0y = [( − ( 0 )2)/20] + 6 = 6
• Make a table of points
 x
 line
 − 12
 − 6
 0
 6
 12
 y
 line
 − [6/5]
 [21/5]
 6
 [21/5]
 − [6/5]
• Graph and find x intercepts

x = − 11,11
Solve (x + 1)2 = − 3 by graphing
• Find standard form ax2 + bc + cy = x2 + 2x + 4
• Find the vertex using x = − [b/2a]x = − ( [2/2(1)] ) = − 1y = ( − 1 )2 + 2( − 1 ) + 4 = 3
• Make a table of points
 x
 line
 − 3
 − 2
 − 1
 0
 1
 y
 line
 7
 4
 3
 4
 7
• Graph and find x intercepts

No solutions
Solve (4 − x) = [9/x] by graphing
• Find standard form ax2 + bc + cy = − x2 + 4x − 9
• Find the vertex using x = − [b/2a]x = − ( [4/(2( − 1))] ) = 2y = − ( 2 )2 + ( 2 )x − 9 = − 5
• Make a table of points
 x
 line
 0
 1
 2
 3
 4
 y
 line
 − 9
 − 6
 − 5
 − 3
 − 9
• Graph and find x intercepts

No solutions
Solve − (x + 8)2 = − 4 by graphing
• Find standard form ax2 + bc + cy = − x2 − 16x − 60
• Find the vertex using x = − [b/2a]x = − ( [( − 16)/(2( − 1))] ) = − 8y = − ( − 8 )2 − 16( − 8 ) − 60 = 4
• Make a table of points
 x
 line
 − 10
 − 9
 − 8
 − 7
 − 6
 y
 line
 0
 3
 4
 3
 0
• Graph and find x intercepts

x = − 10, − 6
Solve x(4x − 28) = − 45 by graphing and estimating
• Find standard form ax2 + bx + cy = 4x2 − 28x + 45
• Find the vertex using x = − [b/2a]x = − ( [( − 28)/2(4)] ) = [7/2]y = 4( [7/2] )2 − 28( [7/2] ) + 45 = − 4
• Make a table of points
 x
 line
 2
 3
 [7/2]
 4
 5
 y
 line
 5
 − 3
 − 4
 − 3
 5

x >> 2[1/2],4[1/2]
Solve [(x2)/9] = 3[3/4] − [x/3] by graphing
• Find standard form ax2 + bc + cy = [(x2)/9] + [x/3] − 3[3/4]
• Find the vertex using x = − [b/2a]x = − ( [([1/3])/(2([1/9]))] ) = − [3/2]y = [(( − [3/2] )2)/9] + [(( − [3/2] ))/3] − 3[3/4] = − 4
• Make a table of points
 x
 line
 − 8[1/2]
 − 4[1/2]
 − [3/2]
 1[1/2]
 5[1/2]
 y
 line
 1[4/9]
 − 3
 − 4
 − 3
 1[4/9]
• Graph and find x intercepts

x ≈ − 7[1/2],4[1/2]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Solving Equations by Graphing

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Solving a Quadratic Equation 0:08
• Example
• Two Distinct Solutions/Roots 8:10
• Roots
• Example: Graphs
• One Double Root 9:19
• Example: One X-Intercept
• No Real Roots 14:03
• Example
• Estimating Solutions 18:41
• Example: Not Integers
• Example 1: Solve by Graphing 20:18
• Example 2: Solve by Graphing 26:36
• Example 3: Solve by Graphing 30:18
• Example 4: Estimate by Graphing 34:59