Grade 10 Mathematics:

Standard Form

Key Points of the Unit:

-how to complete the square

-how to complete the square in a word problem

-how to use the quadratic formula

-quadratic formula in word problems

-how to graph using completing the square and quadratic formula.

Completing the square:

-Vertex form of a quadratic is y = a(x-h)2+k, where (h,k) is the vertex

-Standard form of a quadratic is y = ax2+bx+c

-Since we have the sum (x+3) being squared, we can use the formula for the square of a sum to save time

-The formula for the square of a sum says that (a+b)2 = a2+2ab+b2

-For the formula, a = x and b = 3, so we get x2+6x+9 when we plug inUse the Distributive

Property to distribute the 2 through the parentheses

-y = 2x2+12x-4 is our equation in standard form

Completing the square

Completing the Square With a Word Problem.

Example 1: The height of the Peace Tower is 90m. The path of an object thrown from the Peace Tower can be modelled by the equation:

h =−5d2 +40d+90

where h is the height in metres and d is the horizontal distance in metres. At what horizontal distance did the object reach its maximum height?





h=-5d2 +40d +90

h=-5(d2 -8d) +90

h=-5(d2-8d +16 -16) +90

h=-5(d2-8d +16) +80 +90

h=-5(d-4)2 +170


At 4 meteres, it reaches the max height.

The Quadratic Formula:

The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients" of the quadratic equation they've given you to solve. The Quadratic Formula is derived from the process of completing the square, and is formally stated as:


    For ax2 + bx + c = 0, the value of x is given by


    x= -b +_ (b2 -4ac)

    -----------------------

    2a



The "solution" or "roots" or "zeroes" of a quadratic are usually required to be in the "exact" form of the answer. In the example above, the exact form is the one with the square roots of ten in it. You'll need to get a calculator approximation in order to graph the x-intercepts or to simplify the final answer in a word problem. But unless you have a good reason to think that the answer is supposed to be a rounded answer, always go with the exact form.

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Quadratic formula with decimal

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More Word Problems Using Quadratic Equations - Example 1

Solving a Linear graph using quadratic Formula and Completing the square

to find the x-intercepts, you would need to use the quadratic formula and plot all the numbers down in standard form. At the end, you will get 2 x-intercepts. Find the mid point of those two intercept and than you can solve for the vertex using completing the square method.




Learning Goals:

-I learned how to complete the square

-I learned how to use the quadratic formula

-I learned how to solve word problems using quadratic formula and completing the square

-I learned how to find the vertex and x-intercept using quadratic formula and x-intercept.