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Quadratic Equations

After learning about multiplying binomials using the FOIL method and discussing polynomials, we can move on to working with quadratic equations.

Constructing a Quadratic Equation

We discovered that a polynomial is a product of two or more binomials. Let's consider the simple equality

If two quantities multiply to get zero, does a = 0 or b = 0? or both?

What about the following equality

where we substituted a = (x-2) and b = (x+3). Does x - 2 = 0 or x + 3 = 0, or do they both equal zero?

Let's recall the Zero Product Property , which states that when two quantities multiply to get zero, either one or both of the quantities must be zero.

Because of this property, we can break the product into two equations and solve for x.

Solving for x, we get two different values : x = 2 and x = -3

So we have our solutions, but what does this mean? Observing the equations, we can see that they can be worked with as two seperate linear equations. Let's set both of these equations equal to y and display them on the graph to see if we can see what is going on when we multiply them together.

We can see where the lines hit the x axis (where y equals 0) at -3 and 2. Now let's graph the function when we multiply these linear equations together.

Using the FOIL method of multiplying two binomials, we obtain the function

We have a polynomial function with degree two, which is called a quadratic function. Using the old fasion method of plugging in x values, we can obtain their corresponding y values and graph the function.

Plotting the points, we obtain the image of a parabola.

All quadratic equations form the image of a parabola. Looking at the curve in relation to the linear equations, we can make some conjectures about their relationship.

First, notice that the x intercepts of the linear equations are also the x intercepts of the quadratic function. This makes sense, because the quadratic function is the product of both linear equations. Alone, the linear equations have one value which makes them 0 - but the quadratic function (their product) is 0 either when x is -3 or 2.

Observing the y intercepts, we can see that the y intercept of the the linear equation y = x + 3 is y = 3, the y intercept of the equation y = x - 2 is y = -2, and the y intercept of the quadratic function is y = -6. There is a relationship between them. In fact, the y intercept of the parabola is the product of the y intercepts of the linear equations!

In fact, we can extend this even further. The product of the y value of both linear equations at any given x value will yield y value for the quadratic function! This is why the x intercepts are the same, because if one y value equals zero, than the quadratic function's y value at that x value will be 0.

Analyzing the graph a bit further, we can make a few more conjectures. Interestingly enough, the vertex of the parabola (where the parabola switches directions or where the vertical line of symmetry touches the parabola) is exactly where the distance between the two linear equations is the same from the x axis. Here is a graph illustrating these conjectures

Relating Linear and Quadratic Equations

  • A quadratic can be represented as a product of linear expressions.
  • A quadratic function has the same x intercepts as its linear products.
  • A quadratic function's y intercept is the product of its linear components' y intercepts.
  • A quadratic function's y value anywhere on the graph can be found by the product of y values of its linear components.
  • If the product or y values of the linear equations is positive, the y value of the quadratic will be positive. If the product of y values of the linear equations is negative, the y value of the quadratic will be negative.
  • A quadratic function's vertex is at the point in between the x intercepts where: if the parabola is pointing upwards is the lowest point, if the parabola is opening downwards is the highest point.

Using these conjectures, let's try to construct a parabola given two lines.

Looking at the x intercepts of both lines, we know that the parabola will also have those x intercepts.

Multiplying the y intercepts : 2 * -3, we get the y intercept of the parabola : -6.

We also know that the vertex is located exactly in between the x intercepts. We can draw a vertical line to make it look a little more clear.

Can you see the parabola yet? We know it goes through both x intercepts, and since the y intercept is below the x intercepts, the parabola must open downward. We also know that the vertex is exactly in between both x intercepts. We can find the y value of the vertex by multiplying y values of the linear components at the x value where the line of symmetry hits the x axis. Once we have the y value of our vertex, we can sketch the parabola.

We have constructed the image of a parabola given two lines, and we did it without the equations of the lines. This method of constructing the parabola from two lines isn't very popular and there are definitely easier ways of graphing quadratic functions, but it is beneficial to do this activity to get acquainted with quadratic functions and how they relate to their linear components.

Quadratic Expressions

A quadratic expression is defined as a polynomial of degree 2, which means that the leading term has a variable with an exponent of 2.

A quadratic polynomial can also be given in a factored form as a product of two binomials. We have seen that they can be represented as a product of linear expressions.

For example:

When these expressions are expanded, they take on the General Form of quadratics:

When quadratic expressions are defined as a function, they can take on two different forms - General and Vertex Form.

Forms of Quadratic Functions

General Form

The general form for a quadratic function is given as:

When the quadratic expression is equated with 0, it is then a quadratic equation.

This is the most common form quadratic equations will take on as we work with them, and it is also the simplified form after multiplying two binomials together. In this form, the expression is quadratic if and only if a does not equal 0. If a does equal 0, the expression then becomes linear.

Perhaps the most important aspect of having the general form with the letter coefficients in front of the variables is finding the roots, or x intercepts, of the equation by using the quadratic formula.

In General Form, the coefficient of a determines if the parabola opens up or down. If the coefficient a is positive, the parabola will open upwards. If it is negative, it will open downwards.

Vertex Form

Quadratic Functions can also be written in Vertex Form, which let's algebraically represent where the vertex of the parabola is located. The vertex is also either the minimum or the maximum of the parabola, depending on if it opens upwards or downwards.

The point (x,y) of the vertex is given by (h,k) in the equation

Where h and k can be written in terms of the coefficients a,b, and c of the General Form

Given this quadratic function in general form, find the coordinate of its vertex and rewrite the function in vertex form.

We plug the general form coefficients into our equations for h and k

The vertex of the quadratic funtion is (1,3).

Factored Form

The factored form of a quadratic function clearly gives the roots, or x intercepts of the equation.

where x1 and x2 are the x intercepts. This form also gives us the product of the quadratic's linear components as we have seen before.

Graph the quadratic function in factored form

Finding the Roots of Quadratic Equations

When quadratic functions are equated to 0, they are then considered quadratic equations.

Solving quadratic equations is carried out by finding the roots of the equations, which are the x intercepts of the parabola. This is where the parabola hits the x axis.

where m and n are considered the roots of the equation at (m,0) and (n,0).

There are many different ways of finding the roots of a quadratic equation. Some are more useful than others given the information that we have.

Graphing

Given the graph, we can look at where the parabola touches the x axis.

We can see in this graph that the parabola touches the x axis in two places: (-2,0) and (3,0). Therefore, the roots of the equation are -2 and 3. The equation can then be written as a product.

Sometimes, it is not that easy to see where the the roots are located. We can also solve for the roots algebraically.

The Quadratic Formula

The quadratic formula can be used to solve for the roots given the general form of any quadratic equation. It can be also used to determine how many roots the quadratic equation has.

The Fundamental Theorem of Quadratic Polynomials

Suppose m and n are numbers such that

The sum of m and n is 18 and the product is 10. Solving the second equation for m and subsituting back into the first equation, we get

We can multiply both sides by n and put all the terms on one side to get

Notice that this looks like a quadratic equation in the general form. This is not a coincidence. The solutions, or roots, of this quadratic equation will be the two numbers whose sum is 10 and product is 18. This result is called Viete's Theorem, which is most commonly known as The Fundamental Theorem of Quadratic Polynomials.

*If m and n are solutions to x2 + bx + c = 0

then

In any quadratic polynomial whose leading term has a coefficient of 1, the sum of the roots is the negative of the coefficient of the second term (the x term) and the product of the roots is the last term (the constant term).

In other words, given the general form of a quadratic equation

if a = 1,

-b = sum of the roots of the quadratic polynomial

c = product of the roots of the quadratic polynomial

which is the same as (keeping in mind that a = 1)

The above relationship is used to find the roots of a quadratic equation using factoring.

Factoring Quadratic Equations

The roots of a quadratic equation can be found by factoring the coefficients of the second and last terms. This only works for certain quadratic polynomials which are easily factorable.

For example, given the polynomial below

finding the roots of the quadratic polynomial by factoring involves finding the factors of b and c. In other words, finding values that when added give b and when multiplied give c. It's not always easy to find such 'nice' quadratic polynomials for which this method is easy to use but learning it is key to understanding other methods of finding roots of polynomials.

For example, given the quadratic equation below

when asked to find the roots using factoring, the result would be

where 1 and 2 are the roots of the equation.

Using the relationship discussed before, you should notice that the sum of the roots is the coefficient of the second term and the product of the roots is the third term, i.e.

Therefore, the given quadratic equation can be thought of as

Since not all quadratic polynomials are easily factorable, there is a need for another method to find the roots of quadratic equations.

Completing the Square

Earlier we talked about square quadratic polynomials of the forms (x + α)(x + α) and how these expand to become of the general form:

The roots of the quadratic polynomial above are - and - from

which means that

The above is easy to solve, by taking the square root of both sides of the equation such that

which means that

but since a quadratic polynomial must have 2 roots, we say that

which is better written as

Finding the roots of a quadratic polynomial by completing the square involves modifying the quadratic polynomial such that it becomes a square quadratic polynomial. For example, given the following quadratic equation

The first step in changing the above into a square quadratic equation is moving the last term to the other side of the equal sign. In this case it involves adding 12 to each side:

The second step is to add the square of half of the coefficient of the second term to both sides of equation. In this case you would add 4:

The equation on the left hand side is now a square quadratic equation. (Adding or subtracting the same number from both sides of an equation mathematically does not change the equation).

is now of the same form as

and can easily be factored as

Thus we end up with

The third step is to find the square root of both sides of the equation:

to end up with

Whenever you take the square root of a number you always end up with two results; the positive and negative, in this case +4 and -4.

The last step is to solve for x, in this case by subtracting 2 from both sides of the equation.

and

thus x = 2 and x = -6 are the roots of the equation, which is better written as x = {-6,2}.

Quadratic Equations and Geometry

Quadratic Equations come up on various occasions in geometry. The simplest quadratic equation, x2=k, describes the relationship between the sides of a square (x) and it's area (k).

In fact, this is why quadratics have their name. The variable is squared, which in geometry forms a square, a figure with four sides.

We can represent this relationship through the graph of f(x) = x2.

When the side of a square is 1, the area is 1. When the side is 2, the area is 4, and so on. We do not consider the side of the parabola to the left of y axis because we cannot have negative length in geometry.

Other area formulas, such as the area of a circle, can be represented by quadratic equations.

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