Solving Quadratic Equations Using a GDC

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Solving quadratic equations
(Using GDC)
LO: Use the GDC to solve quadratic equations,
either using the solver or graphically
17 December 2024
Quadratic equations
 
The general form of a quadratic equation is
 
Where 
a
, 
b
 and 
c
 are constants and 
a
 ≠ 0.
 
The 
solutions
 of the equation are the values of 
x
 which
make the equation true.
 
We call these the 
roots
 of the equation, and they are also
the zeros of the quadratic expression 
ax
2
 + 
bx
 + 
c
Solution of 
x
2
 = 
k
 
if 
k
 < 0.
 
exists such that
 
Thus the solutions are 
x
 =
 
if 
k
 = 0
 
x
 =  0
 
if 
k
 > 0.
 
There are no real solutions
Many quadratic equations can be rearranged into the form
 
 
Example 1
:
 
x
2
 –  81 = 0
 
x
2
 =  81
 
x
 =
 
 
x
 =  9
 
x
 =  -9
 
or
 
 
Example 2
:
 
3
x
2
 – 1 = 8
 
3
x
2
 =  9
 
x
 =
 
 
or
 
x
2
 =  3
 
 
Example 3
:
 
(
x 
– 4)
2
  = 6
 
x
 =
 
 
or
 
 
The Null Factor Law
 
The Null Factor Law states:
 
If 
ab
 = 0 then
 a 
= 0 or 
b
 = 0.
 
Solve for x using the Null Factor Law
 
3
x
(
x
 – 5) = 0.
 
 3
x
 = 0
 
 or x
 – 5 = 0
 
 
x
 = 0   
or   
x
 
= 5
 
Solve for x using the Null Factor Law
 
(
x – 4
)(3
x
 + 7) = 0.
 
 
x – 4 
 = 0
 
 or 
3
x
 + 7 = 0
 
 
x
 = 4   
or   3
x
 
= 
7
 
 
Using the GDC
:
 
Solve the quadratic equation:
 
 
Press APPS
 
 
Using the GDC
:
Solve the quadratic equation:
 
Press APPS
 
Press 9 PolySmlt2
 
 
Using the GDC
:
Solve the quadratic equation:
 
Press APPS
Press 9 PolySmlt2
 
Press 1 Polynomial root finder
 
 
Using the GDC
:
Solve the quadratic equation:
 
 
ORDER Press 2
 
 
Press APPS
Press 9 PolySmlt2
Press 1 Polynomial root finder
 
Press F5 NEXT
Using the GDC
:
Solve the quadratic equation:
 
 
 
 
Enter the coefficients
 
Press F5 SOLVE to solve the
equation
ORDER Press 2
Press APPS
Press 9 PolySmlt2
Press 1 Polynomial root finder
Press F5 NEXT
 
3
 
Enter
 
11
 
2
 
Enter
 
Enter
 
Enter
 
Enter
 
+
 
Using the GDC
:
Solve the quadratic equation:
 
 
You have the two solutions
 
 
Press F5 SOLVE to solve the
equation
Enter the coefficients
ORDER Press 2
Press APPS
Press 9 PolySmlt2
Press 1 Polynomial root finder
Press F5 NEXT
3
Enter
11
2
Enter
Enter
Enter
Enter
+
Using the GDC
:
 
Solve the quadratic equation:
 
 
Expanding brackets
 
Collecting like terms
 
Making the RHS zero
 
You have the two solutions
 
3
x – 
3
 + x
2
 + 2
x
 = 3
 
x
2
 + 5
x
 – 3 = 3
 
x
2
 + 5
x
 – 6 = 0
 
 
 
Enter the coefficients
 
Press F5 SOLV to solve the
equation
 
Press F3 COEFF
 
1
 
Enter
 
6
 
5
 
Enter
 
Enter
 
Enter
 
Enter
 
+
 
Using the GDC
:
 
Solve the quadratic equation:
 
 
Multiplying both sides by 
x
 
Expanding the brackets
 
Making the RHS zero
 
3
x
2
 + 2 = 
7x
 
3
x
2
 + 7
x
 + 2 = 0
 
 
 
You have the two solutions
 
Enter the coefficients
 
Press F5 SOLV to solve the
equation
 
Press F3 COEFF
 
3
 
Enter
 
2
 
7
 
Enter
 
Enter
 
Enter
 
Enter
 
+
 
+
Using the GDC
:
 
Use a graphical method to solve:
 
 
We graph
 
On the same set of axes
 
And find where the graphs
intersect
 
Y
1
 = 
8
x
2
 + 
x
 – 2
 
Y
2
 = 
9
 
and
 
 
Using the GDC
:
Solve the quadratic equation:
 
 
Press Y =
 
 
 
Press 2
nd
 QUIT
Using the GDC
:
Solve the quadratic equation:
 
 
Store 
8
x
2
 + 
x
 – 2 
into 
Y1
 and
        
  9
 into 
Y2
.
 
 
 
Press F5 (GRAPH) to draw a graph
of the functions
Press Y =
Press 2
nd
 QUIT
Using the GDC
:
Solve the quadratic equation:
 
 
 
 
Press 2nd F4 (Calc)
Store 
8
x
2
 + 
x
 – 2 
into 
Y1
 and
        
  9
 into 
Y2
.
Press F5 (GRAPH) to draw a graph
of the functions
Press Y =
Press 2
nd
 QUIT
 
Use the arrows to put the cursor on the
line at the left of the 1
st
 intersection.
Enter. Then move the cursor to the
parabola close the intersecting point.
 
Press 5 (Intersect)
 
enter
 
enter
Using the GDC
:
Solve the quadratic equation:
 
 
We have found the first 
point of
intersection 
x
 = -1.24
 
 
 
enter
 
Press 2nd F4 (Calc)
 
Use the arrows to put the cursor on the
line at the left of the 2
nd
  intersection.
Enter. Then move the cursor to the
parabola close the intersecting point.
 
Press 5 (Intersect)
 
enter
 
enter
Using the GDC
:
Solve the quadratic equation:
 
 
 
 
We have found 
the second point of
intersection 
x
 = 1.11
enter
Press 2nd F4 (Calc)
Use the arrows to put the cursor on the
line at the left of the 2
nd
  intersection.
Enter. Then move the cursor to the
parabola close the intersecting point.
Press 5 (Intersect)
enter
enter
We have found the first 
point of
intersection 
x
 = – 1.24
 
x
 = – 1.24
 
x
 = 1.11
 
Solution:
 
or
Thank you for using resources from
https://www.mathssupport.org
If you have a special request, drop us an email
info@mathssupport.org
 
 
For more resources visit our website
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Understanding quadratic equations and their solutions using a graphing calculator. Learn about the general form of quadratic equations, how to rearrange them, and apply the Null Factor Law to solve for roots. Utilize examples and step-by-step instructions to solve quadratic equations using a GDC, including a demonstration for solving the equation 3x^2 + 2x - 11 = 0.


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  1. 17 December 2024 Solving quadratic equations (Using GDC) LO: Use the GDC to solve quadratic equations, either using the solver or graphically www.mathssupport.org

  2. Quadratic equations The general form of a quadratic equation is ax2 + bx + c = 0 Where a, b and c are constants and a 0. The solutions of the equation are the values of x which make the equation true. We call these the roots of the equation, and they are also the zeros of the quadratic expression ax2 + bx + c www.mathssupport.org

  3. Solution of x2 = k Many quadratic equations can be rearranged into the form x2 = k if k > 0. ?= ? ?= ? ? exists such that ? ? Thus the solutions are x = ? x = 0 if k = 0 There are no real solutions if k < 0. www.mathssupport.org

  4. Example 1: x2 81 = 0 x2 = 81 x = x = 9 or x = -9 www.mathssupport.org

  5. Example 2: 3x2 1 = 8 3x2 = 9 x2 = 3 x = ? or x = ? x = ? www.mathssupport.org

  6. Example 3: (x 4)2 = 6 (? 4)2 = 6 x 4 = 6 x = 4 ? or x = 4+ ? x = ? ? www.mathssupport.org

  7. The Null Factor Law The Null Factor Law states: When a product of two (or more) numbers is zero then at least one of them must be zero. If ab = 0 then a = 0 or b = 0. Solve for x using the Null Factor Law 3x(x 5) = 0. 3x = 0 x = 0 or x = 5 or x 5 = 0 (x 4)(3x + 7) = 0. x 4 = 0 or 3x + 7 = 0 x = 4 or 3x = 7 Solve for x using the Null Factor Law x = -? ? www.mathssupport.org

  8. Using the GDC: Solve the quadratic equation: 3x2 + 2x 11 = 0 Press APPS www.mathssupport.org

  9. Using the GDC: Solve the quadratic equation: 3x2 + 2x 11 = 0 Press APPS Press 9 PolySmlt2 www.mathssupport.org

  10. Using the GDC: Solve the quadratic equation: 3x2 + 2x 11 = 0 Press APPS Press 9 PolySmlt2 Press 1 Polynomial root finder www.mathssupport.org

  11. Using the GDC: Solve the quadratic equation: 3x2 + 2x 11 = 0 Press APPS Press 9 PolySmlt2 Press 1 Polynomial root finder ORDER Press 2 Press F5 NEXT www.mathssupport.org

  12. Using the GDC: Solve the quadratic equation: 3x2 + 2x 11 = 0 Press APPS Press 9 PolySmlt2 Press 1 Polynomial root finder ORDER Press 2 Press F5 NEXT 3 Enter 11 Enter Enter the coefficients 2 Enter + Enter Enter Press F5 SOLVE to solve the equation www.mathssupport.org

  13. Using the GDC: Solve the quadratic equation: 3x2 + 2x 11 = 0 Press APPS Press 9 PolySmlt2 Press 1 Polynomial root finder ORDER Press 2 Press F5 NEXT 3 Enter 11 Enter Enter the coefficients 2 Enter + Enter Enter Press F5 SOLVE to solve the equation You have the two solutions www.mathssupport.org

  14. Using the GDC: Solve the quadratic equation: 3(x 1) + x(x + 2) = 3 Expanding brackets 3x 3 + x2 + 2x = 3 Collecting like terms x2 + 5x 3 = 3 x2 + 5x 6 = 0 Press F3 COEFF Making the RHS zero Enter the coefficients 5 Enter + 1 Enter Enter Enter 6 Enter Press F5 SOLV to solve the equation You have the two solutions www.mathssupport.org

  15. Using the GDC: Solve the quadratic equation: ?? +? ?= ? x ?? +? 3x2 + 2 = 7x 3x2 + 7x + 2 = 0 Press F3 COEFF Multiplying both sides by x = ?? ? Expanding the brackets Making the RHS zero Enter the coefficients 7 Enter + 3 Enter Enter + Enter 2 Enter Press F5 SOLV to solve the equation You have the two solutions www.mathssupport.org

  16. Using the GDC: Use a graphical method to solve: 8x2 + x 2 = 9 Y1 = 8x2 + x 2 and Y2 = 9 We graph On the same set of axes And find where the graphs intersect www.mathssupport.org

  17. Using the GDC: Solve the quadratic equation: 8x2 + x 2 = 9 Press 2nd QUIT Press Y = www.mathssupport.org

  18. Using the GDC: Solve the quadratic equation: 8x2 + x 2 = 9 Press 2nd QUIT Press Y = Store 8x2 + x 2 into Y1 and 9 into Y2. Press F5 (GRAPH) to draw a graph of the functions www.mathssupport.org

  19. Using the GDC: Solve the quadratic equation: 8x2 + x 2 = 9 Press 2nd QUIT Press Y = Store 8x2 + x 2 into Y1 and 9 into Y2. Press F5 (GRAPH) to draw a graph of the functions Press 2nd F4 (Calc) Use the arrows to put the cursor on the line at the left of the 1st intersection. Enter. Then move the cursor to the parabola close the intersecting point. enter enter Press 5 (Intersect) www.mathssupport.org

  20. Using the GDC: Solve the quadratic equation: 8x2 + x 2 = 9 We have found the first point of intersection x = -1.24 enter Press 2nd F4 (Calc) Use the arrows to put the cursor on the line at the left of the 2nd intersection. Enter. Then move the cursor to the parabola close the intersecting point. enter enter Press 5 (Intersect) www.mathssupport.org

  21. Using the GDC: Solve the quadratic equation: 8x2 + x 2 = 9 We have found the first point of intersection x = 1.24 enter Press 2nd F4 (Calc) Use the arrows to put the cursor on the line at the left of the 2nd intersection. Enter. Then move the cursor to the parabola close the intersecting point. enter enter Press 5 (Intersect) We have found the second point of intersection x = 1.11 Solution: x = 1.24 or x = 1.11 www.mathssupport.org

  22. Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org www.mathssupport.org

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