All About Quadratics

Standard Form

By: Suhasee Patel

Learning goal 1

The standard form equation tells us about which way the parabola is going to open and the axis of symmetry by using the formula: -b/2a



    • If, a>0 the parabola opens upwards
    • if a<0 it opens downwards
    • The axis of symmetry is the line x=-b/2a

Learning goal 2

The quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way.


Here x represents an unknown, while a, b, c and are constants with a not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation, by inserting the former into the latter. Each of the solutions given by the quadratic formula is called a root of the quadratic equation.

Geometrically, these roots represent the x values at which any parabola, explicitly given as y=ax^2+bx+c crosses the -axis.

Learning Goal 3

When turning an equation from standard form to vertex form you use a technique called completing the square.


For Example:

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Summary

Completing The Square

The process of completing the square includes changing the first two terms of the quadratic relation of the form y=ax^2 + bx + c and make it into a perfect square.
❖ Completing the Square - Solving Quadratic Equations ❖

Quadratic Formula

All quadratic equations of the form ax²+bx+c,can be solved using the quadratic formula. It is used by substituting the values of a, b and c from the original equation, into the formula, then solving. ​ ​

By using this formula it is easier to solve for the solutions or the roots of the quadratic equation. This is a another way to solve an equation if it cannot be factored.


Here are the steps on how to solve the formula:

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Solving Quadratic Equations using the Quadratic Formula - Example 1

Graphing Standard Form

Steps:
  1. Find the x-intercepts of the equation, by using the quadratic formula.
  2. Find the vertex of the equation, by completing the square.
  3. Graph the x-intercepts and the vertex, then draw the parabola.

Discriminant

Discriminant allows you to see how many solutions that the quadratic equations will have without having to use the whole quadratic formula to solve it. In order to find the discriminant the formula would be the same as the quadratic formula but we do not have to square it or divide it. So therefore, the formula would be b^2 -4ac

Word Problems

Revenue Word Problems:

Revenue = (Current price +/- Price decreased/increased X) (Current sales parameters +/- Number parameters decreased)

  • For example, If Bill Gates sells 15 copies a day of Windows 7 at the current price of $299, he would sell 4 more copies of Windows 7 per day if the price decreases $5.
    • X will be the amount of times increased in the price.
    • Equation would be: Revenue = (299-5X)(15 + 4X)
    • The amount you increase will be X.


For Example:

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Geometry Word Problems:


  • Helps us find the unknown lengths of the shape.
  • We can use the quadratic formula in order to solve it
  • Most of the time when we solve it we will get a positive and a negative number, so we should always use the positive number to sub into the the original equation because the length can never be negative!


For Example:

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Reflection

Overall, the quadratics unit was very easy and once you had begun to actually understand the concepts it became much easier to practice them and perfect them.


The Standard Form unit test was the test that I did fairly well on, and many people had said that it becomes easy once you understand it. I also figured out that once you begin to understand vertex form and factored form standard form becomes much easier. The unit that I had some difficulty on was factoring because at first it was hard to remember the different ways to factor but after constantly doing my homework it began to make more sense to me.


Some connections that I have made throughout the quadratics unit is how when we factor the equations it can help us graph the parabola and find the vertex, x-intercepts and y-intercepts. Also if we use the quadratic formula we can solve for x and then we can graph the equation while in vertex form.

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