Download - 4.6 Q UADRATIC EQUATION AND THE D ISCRIMINANT. Q UIZ : S OLVE BY USING THE QUADRATIC FORMULA
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4.6 QUADRATIC EQUATION AND THE DISCRIMINANT
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QUIZ: SOLVE BY USING THE QUADRATIC FORMULA
Y = ax² + bx + c
Y = x² + 3x + 2
a = 1, b = 3, c = 2
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CAN YOU SIMPLIFY THE FORMULA?
Y = ax² + bx + cY = x² + 3x + 2
x = , x =
x = - 1 , x = -2
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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
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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
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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 = -
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YOUR TURN:
Y = ax² + bx + c 0 = ax² + bx + c
Y = x² - 4x + 3
a = 1, b = -4 , c = 3
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= ±
= 3 = 1
x = 3, 1
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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
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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?
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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?
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14. WHICH ONE OF THE FOLLOWING EQUATIONS WILL HAVE IMAGINARY SOLUTIONS?
y = x² + x + 5 y = x² - x - 5
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15. WHICH ONE OF THE FOLLOWING EQUATIONS HAS ONLY ONE X-INTERCEPT?
y = x² + 6x + 9 y = x² + 4x + 9
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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
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YOUR TURN:
16. Write the part of the quadratic that is the discriminate.
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WHAT HAPPENS IF THE DISCRIMINATE IS A POSITIVE NUMBER?
,
x = , x =
x = - 1 , x = -2
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WHAT HAPPENS WHEN THE DISCRIMINATE IS ZERO?
, x = -3
The parabola only touches the x-axis in one place.
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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?
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WHAT HAPPENS IF THE DISCRIMINATE IS NEGATIVE?
x = + , x = - x = 1 ± i
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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)
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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?