Solving by factoring and extracting the square roots are techniques that only work for very special quadratic equations. In this section, we will develop a method that will solve any quadratic equation. This technique will be used to develop the quadratic formula.
is to be a perfect square then the constant term must equal 16. In this case, we can see that
To know 16 was the number needed to complete the square we divided our middle term B = 8 by 2 to get 4 then squared that to obtain 16.
Complete the square then factor.
The leading coefficient must be 1 if this method is to work. If it is not equal to 1 we could always factor the coefficient out first as in the next examples.
Determine the constant that completes the square.
Now that you know how to complete the square for a quadratic expression you are ready to solve any quadratic equation. Notice that the next example does not factor, so we will solve it by completing the square.
Solve by completing the square:
Step 1: Add or subtract the constant term to the other side.
Step 2: Add (b/2)^2 to both sides. Here B = −10 so (−10/2)^2 = (−5)^2 = 25
Step 3: Factor and then apply the square root property.
Step 4: Solve as usual.
Tip: Be sure to add the “magic number,”
to both sides so that the equality is retained.
Solve by extracting roots.
So far we have seen examples where the leading coefficients are 1. But if this technique is to be useful for solving quadratic equations of the form
we must address the case where a is a number other than 1.
Solve by completing the square.
Extra Step: Factor out the leading coefficient.
Here B = −6 so (B/2)^2 = (−6/2)^2 = (−3)^2 = 9. Caution! Adding 9 inside the parentheses is equivalent to adding 18 here.
An ellipse is the set of all points whose distances to two fixed points add to the same constant. In this section, we will focus on the equation of ellipses.
The long axis of an ellipse is called the major axis and the short axis is called the minor axis. These axes terminate at points that we will call vertices. The vertices along the horizontal axis will be (h ± a,k) and the vertices along the vertical axes will be (h, k ± b). These points, along with the center, will provide us with a method to sketch an ellipse given standard form.
Given the equation of an ellipse in standard form we can graph it using the following steps:
First plot the center (h, k).
Plot a point a units up and down from the center.
Plot a point b units left and right of the center.
Sketch the ellipse and label the four points.
In this example the major axis is the vertical axis and the minor axis is the horizontal axis. The major axis measures 2b = 10 units in length and the minor axis measures 2a = 6 units in length. There are no x- and y- intercepts in this example.
Graph the ellipse. Label the center and 4 other points.
It will often be the case that the ellipse is not given in standard form. In this case, we will begin by rewriting it in standard form.
Graph the ellipse. Label the center and four other points.
The next set of examples give the equations in general form. Complete the squares to obtain standard form. Remember to factor the leading coefficient out of each variable grouping before using (B/2)^2 to complete the square.
Graph the ellipse. Label the center and four other points.
Answer the questions.
When the calculation for intercepts yields complex results then there are no real intercepts. This just means that the graph does not cross that particular axis.
Example: Find the equation of the ellipse centered at (3, -1) with a horizontal major axis of length 10 units and vertical minor axis of length 4 units.
Example: Find the equation of the ellipse given the vertices ( ±3,0) and (0, ±8).
Example: Find the equation of the ellipse whose major axis has vertices (-1, -2) and (-1, 10) and minor axis has co-vertices (-3, 4) and (1, 4).
