It has become somewhat fashionable to have students derive the Quadratic Formula themselves; this is done by completing the square for the generic quadratic equation ax2 + bx + c = 0. While I can understand the impulse (showing students how the Formula was invented, and thereby providing a concrete example of the usefulness of abstract symbolic manipulation), the computations involved are often a bit beyond the average student at this point.
Answer:
Circumcenter Incenter Centroid
Formed by intersection of Perp. Bisectors Angle bisectors Medians
Type of circle Circumscribed Inscribed No circle
Special property Equidistant from Equidistant from Center of mass
vertices Sides
The vertex form of a quadratic function is:
f(x) = a(x - h)² + k
The coordinate (h, k) represents a parabola's vertex.
In order to convert a quadratic function in standard form to the vertex form, we can complete the square.
y = 2x² - 5x + 13
Move the constant, 13, to the other side of the equation by subtracting it from both sides of the equation.
y - 13 = 2x² - 5x
Factor out 2 on the right side of the equation.
y - 13 = 2(x² - 2.5x)
Add (b/2)² to both sides of the equation, but remember that since we factored 2 out on the right side of the equation we have to multiply (b/2)² by 2 again on the left side.
y - 13 + 2(2.5/2)² = 2(x² - 2.5x + (2.5/2)²)
y - 13 + 3.125 = 2(x² - 2.5x + 1.5625)
Add the constants on the left and factor the expression on the right to a perfect square.
y - 9.875 = 2(x - 1.25)²
Now, we need y to be by itself again so add 9.875 back to both sides of the equation to move it back to the right side.
y = 2(x - 1.25)² + 9.875
Vertex: (1.25, 9.875)
Solution: y = 2(x - 1.25)² + 9.875
Or if you prefer fractions
y = 2(x - 5/4)² + 79/8
Key info-there are nine boys for every six girls on the softball team
9/6
9(3)=27. 6(3)18
18+27=45 5 more to go
if there is 50 students on each of the teams
9(5)=45
6(8)=48
As one can see,they both cannot multiply up to 50,but they round up.
