4.6 QUADRATIC EQUATION AND THE DISCRIMINANT

QUIZ: SOLVE BY USING THE QUADRATIC FORMULA

Y = ax² + bx + c

Y = x² + 3x + 2

a = 1, b = 3, c = 2

CAN YOU SIMPLIFY THE FORMULA?

Y = ax² + bx + cY = x² + 3x + 2

x = , x =

x = - 1 , x = -2

WHAT IF IT IS NOT IN STANDARD FORM?

2x = 3x² + 5(Hint: Remember what you do to the left you do to the right)

0 = 3x² + 2x + 5

a = ?, b = ? , c = ?

a = 3, b = 2 ,c = 5

YOUR TURN:

Y = ax² + bx + c 0 = ax² + bx + cDefine the values : a = ? , b = ? , c = ?

1. Y = 2 – 12x² + 3

2. 5x = 3x² - 5x + 1

3. Y = 3x + 4x² + 7

THE EQUATIONS CAN RESULT IN MORE COMPLICATED ARITHMETIC

Y = ax² + bx + cY = x² +x - 1 a = 1, b = 1, c = -1 (Simplify)

x = , x =

x = + , x = -

YOUR TURN:

Y = ax² + bx + c 0 = ax² + bx + c

Y = x² - 4x + 3

a = 1, b = -4 , c = 3

= ±

= 3 = 1

x = 3, 1

YOUR TURN:USE THE QUADRATIC EQUATION TO SOLVE

5. y = x² - 166, y = x² - 2x

Which one of the equations is in standard form?a) y = 2 (x – 4)² + 13b) y = 5 ( x – 3)( x – 4)c) y = 5x² + 6x + 8

USE THE QUADRATIC EQUATION TO SOLVE

8. y = x² - 4x + 4 (Solve using the quadratic equation)

9.Why does the equation only have 1 x-intercept?

10. What happened to the quadratic equation to have the equation produce 1 x – intercept?

YOUR TURN: USE THE QUADRATIC TO SOLVE

11. y = x² + 4x + 4 (Use the quadratic equation).

12. What are the x-intercepts of the parabola? Does it cross the x axis

13. What happened in the quadratic equation that made the x-intercepts different? What kind of x-intercepts are they?

14. WHICH ONE OF THE FOLLOWING EQUATIONS WILL HAVE IMAGINARY SOLUTIONS?

y = x² + x + 5 y = x² - x - 5

15. WHICH ONE OF THE FOLLOWING EQUATIONS HAS ONLY ONE X-INTERCEPT?

y = x² + 6x + 9 y = x² + 4x + 9

VOCABULARY: DISCRIMINATE

The discriminate is the part of the quadratic that is under the square root of the quadratic equation. It is the radicand of the formula.

b² - 4ac

YOUR TURN:

16. Write the part of the quadratic that is the discriminate.

WHAT HAPPENS IF THE DISCRIMINATE IS A POSITIVE NUMBER?

,

x = , x =

x = - 1 , x = -2

WHAT HAPPENS WHEN THE DISCRIMINATE IS ZERO?

, x = -3

The parabola only touches the x-axis in one place.

HOW DOES IT ONLY TOUCH THE X-AXIS IN ONLY ONE PLACE

y = x² + 4x + 4 (perfect square trinomial)

Y = ( x + 2 ) ²

Y = ( x + 2 ) ( x + 2 )

0 = ( x + 2) ( x + 2)

X = -2

What does the graph look like?

WHAT HAPPENS IF THE DISCRIMINATE IS NEGATIVE?

x = + , x = - x = 1 ± i

THE DISCRIMINATE IN THE RADICAND OF THE QUADRATIC EQUATION:

b² - 4ac 2 Real solutions ( parabola crosses x-axis at two points)

b² - 4ac 1 Real solution (parabola crosses x-axis at one point)

(The equation was a perfect square trinomial)

b² - 4ac 2 Imaginary solutions (the parabola does not cross the x axis)

(No reals solutions)

YOUR TURN:

Find the discriminate of the following equations:17. y = x² - 2x + 118. y = x² - 2x – 119. y = x² - 4x + 320. How many solutions for problem #17?21. How many solutions for problem # 18?22. How many solution for problem # 19?