2. Quadratic formula
3. Word problems/tips
Summary of the unit
Completing the Square
Step 2: Common factor only the constant terms, if posssible.
Step 3: Complete the square.
Step 4: Write it as a binomial squared.
Example: y= 4x^2 + 16x + 3 y= (4x^2 + 16x) + 3 y= 4(x^2 + 4x) + 3 y= 4(x^2 + 4x + 4 - 4) + 3
-> There needs to be one positive and one negative number to balance each other out
4 ^2 2 y= 4[(x^2 + 4x + 4) - 4] + 3 y= 4 (x+2)^2 - 4 + 3 y= 4 (x+2)^2 - 1
The vertex is (-2, -1)
To find the x-intercepts/zeroes in an equation that cannot be factored, you can use the quadratic formula
In order to use this formula, your equation needs to be in standard form - y=ax^2 +bx + c
Substitute the a, b, and c values into the formula to get your answer.
3x^2 - 14x - 5
a = 3, b = -14, c = -5
x = 14 -14^2 - 4(3)(-5)
= 14 256
The exact roots are
14+ 256 and 14 - 256
The approximate roots are 16.67 and 11.33 In the case of a negative number inside the square root, the solution is not possible because you cannot square root a negative number.
Word Problems Tips/Hints
To find the x-intercepts - use the quadratic formula.
A garden measuring 12 metres by 16 metres is to have a pedestrian pathway installed all around it, increasing the total area to 285 square metres. What will be the width of the pathway?
The height of a coin in the air t seconds after it is flipped can be modeled by y = -16t^2 + 32t + 5, where t and y are measured in m. What is the maximum height that the coin reaches?
->To solve this, we need to complete the square because it is asking for the maximum height (y value of the vertex). y=(-16t^2 + 32t) + 5 y= -16(t^2 - 2) + 5 y= -16 (t^2 - 2 + 1) -1) +5 -> -16 multiplies with -1 y= -16 (t-1)^2 + 21
The vertex is (1,21).
Therefore the maximum height of the coin is 21m.