21-22QuadFormCompleteSquare Flashcards
What does the Quadratic formula do?
It finds the x-intercepts when the equation is not easily factored.
How long has the Quadratic formula been around?
800 years
What is the formula?
[-b±√(b² - 4ac)]/2a
where ax² + bx + c = 0
What are the x-intercepts of
x² - 12x - 5 = 0
x² - 12x - 5 = 0
a = 1; b = -12; c = -5
+12±√(144 - -20)/2
12±√164/2
12±2√41 / 2
6±√41
What are the x-intercepts of:
x² + 2x + 5 = 0
x² + 2x + 5 = 0
a = 1; b = 2; c = 5
-2±√(4 - 20)/2
Square root of negative number.
No solution.
It’s above the x-axis.
How do you determine the coordinates of the bottom of a parabola that points up?
Complete the Square
Find the lowest y possible.
How do you complete the square?
x² + 4x - 6 = 0
Take the x coefficient, divide in half and square
(x² + 4x + 4) - 6 - 4 = 0
(x + 2)² - 10 = 0
What is the bottom of the parabola:
f(x) = x² + 6x + 35
f(x) = x² + 6x + 35
(x² + 6x + 9) + 35 - 9
f(x) = (x + 3)² + 26
(x + 3)² cannot be negative so the lowest y = 26.
26 = x² + 6x + 35
-9 = (-3)² + (6*-3)
x = -3
lowest point = (-3, 26)
What is the lowest point on the parabola:
f(x) = x² - 10x - 23
f(x) = x² - 10x - 23
f(x) = (x² - 10x + 25) - 23 - 25
f(x) = (x - 5)² - 48
lowest y = -48
- 48 = x² - 10x - 23
- 25 = x² - 10x
x = 5
What is the lowest point on the parabola:
x² + 5x - 24 = 0
x² + 5x - 24 = 0
(x² + 5x + 25/4) - 24 - 25/4 = 0
(x² + 5x + 25/4) - 30 1/4 = 0
y value = -30 1/4
x-value = -5/2
[y = (x - m)² + n
vertex = (m,n)]
(x + 5/2)² = 30 ¼
x + 5/2 = ±√121 /√4 = ±11 /2
x = 5/2 ±11 /2 = x-intercepts
