+/-/x Polynomials
(x² + 3x + 4) + (2x² - x -5)
x² + 3x + 4 + 2x² - x -5
3x² + 2x - 1
Solve for n.
n² + 9n + 20= 0
(n + 4)(n + 5) = 0
n = -4 or n = -5
Factor.
z² - 25
(z + 5)(z - 5)
How does k change the graph compared to f(x) = x²?
+k = shifts up
- k = shifts down
x² + 3x + 9
no solution
(3x² - 4x + 7) - (2x² + 3 - 9)
3x² - 4x + 7 - 2x² - 3 + 9
x² - 4x + 13
Solve for r.
r² + 16 - 8r = 0
r² - 8r + 16 = 0
(r - 4)(r - 4) = 0
r = 4
Factor.
7f² + 29f + 4
7f² + 29f + 4
7f² + 28f + 1f + 4
7f(f + 4) + 1(f + 4)
(f + 4)(7f + 1)
How does h change the graph compared to f(x) = x²?
+h : shifts right
-h : shifts left
x² + 6x + 4 = 0
x² + 6x =-4
x² + 6x + 9 =-4 + 9
(x + 3)(x + 3) = 5
sqrt ((x + 3)(x + 3)) = +- sqrt(5)
x + 3 = +- sqrt(5)
x = -3 +- sqrt(5)
2(x + 3) - 4(x² - 4)
2x + 6 - 4x²+ 16
- 4x² + 2x + 22
Solve for v.
v² - 8v + 12 = 0
(v - 2)(v - 6) = 0
v = 2 or v = 6
3h² - 17h + 20
3h² - 12h -5h + 20
(3h² - 12h) + (-5h + 20)
3h(h - 4) + -5(h - 4)
(3h - 5)(h - 4)
Write the general equation for the standard form and vertex form of a quadratic equation.
Standard:
ax2 + bx + 3
Vertex:
a(x-h)2 + k
2x² - 5x + 10 = 0
(5+-sqrt5)/2
(x + 3)(x² + 5)
x^3 + 3x^2 + 5x + 15
Factor.
21 - 22m + m²
m² - 22m + 21
(m - 1)(m - 21)
Factor.
25y² - 36
(5y - 6)(5y + 6)
Describe the a, h, and k of the graph of this equation:
2(x-1)²+3
a = More closed. End behaviors point up.
h = shifts right 1 unit
k = shifts up 3 units
3x² - 5x - 7 = 0
(-5+-sqrt(109))/6
-3x(x-6)(x+6)
(-3x2 + 18)(x + 6)
= -3x3 - 18x2 + 18x + 108
Factor.
9 + 10t + t²
t² + 10t + 9
(t + 1)(t + 9)
100a² - 100a + 25
25(4a2 - 4a + 1)
25[4a2 - 2a - 2a + 1]
25[(4a2 - 2a) + (-2a + 1)]
25[2a(2a - 1) + -1(2a - 1)]
25(2a - 1)(2a - 1)
25(2a - 1)2
Describe the a, h, and k of the graph of this equation:
½ (x+3)²-2
a = More open. End behaviors points up.
h = (x-(-3)) = shifts left 3 units
k = shifts down 2 units
When should you complete the square to solve a quadratic equation?
When an equation is not factorable by whole numbers.