© Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology
T3 Scotland
Quadratic Functions
Formula and Discriminant
©Teachers Teaching with Technology (Scotland)
THE QUADRATIC FORMULA
The Quadratic Formula allows us to find the roots of any quadratic equation
ax2 + bx + c from the coefficients a, b and c.
2
If ax + bx + c = 0
- b ± b 2 - 4ac
then x =
, a ≠0
2a
Example
Solve 2x 2 - 5x + 1 = 0
ax2 + bx + c = 0
a = 2, b = -5,
c = 1.
c.f.
1. Store the values of a, b
and c. Using the
[STO] and [ALPHA]
buttons.
6. You can also get the TI-83 to give
the answer correct to two decimal
places by setting the [MODE]
screen to two decimal places.
2. Enter the quadratic
formula, use plenty of
brackets.
3. Pressing [ENTER],
the TI-83 evaluates
the formula and
returns the first root.
7. Using the [2nd][ENTRY] key the
calculator can be made to repeat the
calculations.
This time giving the answers to 2 d.p.
4. Using the
[2nd][ENTRY] key
recall the previous line
and then edit the +
sign to a - sign.
5. Pressing [ENTER], the
TI-83 evaluates the new
formula returns the
second root.
T3 Scotland
x = 2.28 and 0.22 are the roots
of 2x2 - 5x + 1 = 0 to 2 d.p.
Quadratic Function: Formula and Discriminant.
Page 1 of 6
FINDING QUADRATIC ROOTS FROM A GRAPH
Example (same again)
Solve 2x2 - 5x + 1 = 0 rounding the roots to two decimal places.
1. Enter the function on
the [Y=] screen and
[ZOOM]4:ZDecimal
9. You can also get the TI-83 to give
the answer correct to two decimal
places by setting the [MODE]
screen to two decimal places.
2. The TI-83 draws this
graph. The roots (or
zeros) can be seen.
10.
3. To calculate the value
of the roots press
[CALC]2:zero
Now repeating steps 3 to 8 the
TI-83 will calculate the roots
correct to 2 decimal places.
4. The TI-83 prompts you
for a “Left Bound”.
Using the cursor keys
move to just left of the
root you want and
[ENTER]
5. Now a “Right Bound”.
Use the cursor keys and
[ENTER].
x = 2.28 and 0.22 are the roots
of 2x2 - 5x + 1 = 0 to 2 d.p.
6. Now a “Guess”.
Give your best estimate,
using the cursor keys
and [ENTER].
7. The TI-83 returns its
best approximation for
the root.
8. Repeat steps 3 -7.
This time for the other
root. The TI-83 returns
this value.
T3 Scotland
Quadratic Function: Formula and Discriminant.
Page 2 of 6
Exercise 1
Solve these quadratic equations giving the answer to 2 decimal places.
i)
x2 + 4x + 1 = 0
ii)
x2 + 6x + 4 = 0
iii)
x2 + 7x + 5 = 0
iv)
x2 + 2x - 1 = 0
v)
x2 - 6x + 3 = 0
vi)
x2 + 4x + 1 = 0
vii)
x2 = 12x + 5 = 0
viii) x2 + 8x = 10
ix)
2x2 + x - 4 = 0
x)
5x2 + 3x - 4 = 0
xi)
10x2 = 7x + 1 = 0
xii)
10x2 = -12x + 9
xiii) 3·5x2 + 4·2x - 0·32 = 0
T3 Scotland
xiii) 1·6x2 + 2·08x - 2·10 = 0
Quadratic Function: Formula and Discriminant.
Page 3 of 6
THE DISCRIMINANT
The Quadratic Formula allows us to find the roots of any quadratic equation
ax2 + bx + c from the coefficients a, b and c.
But does it always work?
- b ± b 2 - 4ac
then x =
, a≠0
2a
2
If ax + bx + c = 0
Using the method on page 1 or 2, solve these Quadratic Equations and complete the tables.
Describe the nature of the roots as “real & equal” or “real & unequal” or “non-real”.
Sketch each graph in the grid provided.
a
1.
b2 - 4ac
a
b2 - 4ac
a
b
c
Nature of roots
b
c
y = x2 - 4x + 5
roots
b2 - 4ac
a
4.
Nature of roots
y = x2 - 4x + 4
roots
3.
c
y = x2 - 4x + 3
roots
2.
b
Nature of roots
b
c
y = 4x2 - 9x + 2
roots
T3 Scotland
b2 - 4ac
Nature of roots
Quadratic Function: Formula and Discriminant.
Page 4 of 6
a
5.
b2 - 4ac
a
b2 - 4ac
a
b2 - 4ac
a
b2 - 4ac
a
b2 - 4ac
a
b
c
Nature of roots
b
c
Nature of roots
b
c
Nature of roots
b
c
y = 9x2 + 6x + 1
roots
T3 Scotland
Nature of roots
y = x2 - 6x + 9
roots
10.
c
y = x2 + 2x + 3
roots
9.
b
y = x2 - 5
roots
8.
Nature of roots
y = x2 - x - 5
roots
7.
c
y = x2 + 2x + 1
roots
6.
b
b2 - 4ac
Nature of roots
Quadratic Function: Formula and Discriminant.
Page 5 of 6
Summarise the information from pages 4 and 5 on this table .
roots
1.
y = x2 - 4x + 3
2.
y = x2 - 4x + 4
3.
y = x2 - 4x + 5
4.
y = 4x2 - 9x + 2
5.
y = x2 + 2x + 1
6.
y = x2 - x - 5
7.
y = x2 - 5
8.
y = x2 + 2x + 3
9.
y = x2 - 6x + 9
10.
y = 9x2 + 6x + 1
b2 - 4ac
Nature of roots
For Quadratic Equations
ax 2 + bx + c = 0,
b2 - 4ac is called the Discriminant
Use the results summarised above to complete these statements.
If
b2 - 4ac > 0 the roots are _______________________________.
If
b2 - 4ac = 0 the roots are _______________________________.
If
b2 - 4ac < 0 the roots are _______________________________.
T3 Scotland
Quadratic Function: Formula and Discriminant.
Page 6 of 6