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© Teachers Teaching with Technology (Scotland)
©Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology
T3 Scotland
Quadratic Functions
Formula and Discriminant
� T3 Scotland Quadratic Function: Formula and Discriminant. Page 1 of 6
THE QUADRATIC FORMULAThe Quadratic Formula allows us to find the roots of any quadratic equationax2 + bx + c from the coefficients a, b and c.
ExampleSolve 2x2 - 5x + 1 = 0
c.f. ax2 + bx + c = 0a = 2, b = -5, c = 1.
1. Store the values of a, b and c. Using the [STO] and [ALPHA] buttons.
2. Enter the quadratic formula, use plenty of brackets.
3. Pressing [ENTER], the TI-83 evaluates the formula and returns the first root.
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.
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.
7. Using the [2nd][ENTRY] key the calculator can be made to repeat the calculations.This time giving the answers to 2 d.p.
x = 2.28 and 0.22 are the roots of 2x2 - 5x + 1 = 0 to 2 d.p.
If a b c then b b aca
ax x x22
0 42
0+ + = =- ± - , ≠
� T3 Scotland Quadratic Function: Formula and Discriminant. Page 2 of 6
FINDING QUADRATIC ROOTS FROM A GRAPHExample (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
2. The TI-83 draws this graph. The roots (or zeros) can be seen.
3. To calculate the value of the roots press [CALC]2:zero
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].
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.
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.
10. Now repeating steps 3 to 8 the TI-83 will calculate the roots correct to 2 decimal places.
x = 2.28 and 0.22 are the roots of 2x2 - 5x + 1 = 0 to 2 d.p.
T3 Scotland Quadratic Function: Formula and Discriminant. Page 3 of 6
Exercise 1
Solve these quadratic equations giving the answer to 2 decimal places.
i) x2 + 4x + 1 = 0 ii) x2 + 6x + 4 = 0iii) x2 + 7x + 5 = 0 iv) x2 + 2x - 1 = 0v) x2 - 6x + 3 = 0 vi) x2 + 4x + 1 = 0vii) x2 = 12x + 5 = 0 viii) x2 + 8x = 10ix) 2x2 + x - 4 = 0 x) 5x2 + 3x - 4 = 0xi) 10x2 = 7x + 1 = 0 xii) 10x2 = -12x + 9xiii) 3·5x2 + 4·2x - 0·32 = 0 xiii) 1·6x2 + 2·08x - 2·10 = 0
� T3 Scotland Quadratic Function: Formula and Discriminant. Page 4 of 6
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.
THE DISCRIMINANT
The Quadratic Formula allows us to find the roots of any quadratic equationax2 + bx + c from the coefficients a, b and c.
But does it always work?
1. y = x2 - 4x + 3a b c
roots b2 - 4ac Nature of roots
2. y = x2 - 4x + 4a b c
roots b2 - 4ac Nature of roots
3. y = x2 - 4x + 5a b c
roots b2 - 4ac Nature of roots
4. y = 4x2 - 9x + 2a b c
roots b2 - 4ac Nature of roots
If a b c then b b aca
ax x x22
0 42
0+ + = =- ± - ≠,
� T3 Scotland Quadratic Function: Formula and Discriminant. Page 5 of 6
7. y = x2 - 5a b c
roots b2 - 4ac Nature of roots
8. y = x2 + 2x + 3a b c
roots b2 - 4ac Nature of roots
9. y = x2 - 6x + 9a b c
roots b2 - 4ac Nature of roots
10. y = 9x2 + 6x + 1a b c
roots b2 - 4ac Nature of roots
6. y = x2 - x - 5a b c
roots b2 - 4ac Nature of roots
5. y = x2 + 2x + 1a b c
roots b2 - 4ac Nature of roots
� T3 Scotland Quadratic Function: Formula and Discriminant. Page 6 of 6
roots b2 - 4ac
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
Nature of roots
9. y = x2 - 6x + 9
10. y = 9x2 + 6x + 1
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 _______________________________.
Summarise the information from pages 4 and 5 on this table .
For Quadratic Equations
ax2 + bx + c = 0,
b2 - 4ac is called the Discriminant