We are now going to learn how to write an equation in standard form. There are a few different ways in which we can learn how to switch an equation from factored form to standard form. We are going to look at one of those methods: FOIL.
Let's look at two sets of binomials: (x - 3) and (x + 7).
If we put them side by side, they would look like this: (x - 3)(x + 7).
Now, let's look at the two sets a little more closely.
The red terms are the FIRST terms (x-3)(x+7)
The orange terms are the INSIDE terms (x-3)(x+7)
The blue terms are the OUTSIDE terms (x-3)(x+7)
The green terms are the LAST terms (x-3)(x+7)
The letters of FOIL stand for:
First
Outside
Inside
Last
So, we need to multiply the first terms, the ouside terms, the inside terms, and the last terms, and add all of the terms together.
Using the example (x - 3)(x + 7), if we were to multiply them, we would get:
(x - 3)(x + 7)
=(x)(x) + (x)(7) + (-3)(x) + (-3)(7)
= x² + 7x - 3x - 21
= x² + 4x - 21
This is the equation in standard form
Example 1: Expand:
(2x + 5)(7x + 2)
We are going to follow the same procedure as we did earlier. This time, however, it will be a little more complicated because we have coefficients in front of our x terms.
(2x + 5)(7x + 2)
=(2x)(7x) + (2x)(2) + (5)(7x) + (5)(2)
= 14x² + 4x + 35x + 10
= 14x ² + 39x + 10
Example 2: Expand: (x - 5)²
Recall: x² = (x)(x).
Therefore, (x - 5)² = (x - 5)(x - 5)
(x - 5)(x - 5)
= x² - 5x - 5x + 25
= x² - 10x + 25
Example 3: Expand
2(x+5)(x-2)
Would we start by multiplying the 2 through the first bracket, or would we start by using FOIL for the second two brackets?
Either way you chose, you were right! It does not matter which method you choose. However, it may be easier to use FOIL first, because then you will be working with smaller numbers.
2(x + 5)(x - 2)
=2 (x² - 2x + 5x + 10)
= 2 (x² + 3x + 10)
= 2 x ² + 6x + 20
That is your final answer.
Example 4
Find the equation of the parabola, in standard form, if the roots are -5 and 3, and a y-intercept of -30.
RECALL: In factored form we would write y = a(x - s)(x - t).
Since the y-intercept is -30, we can substitute (0,30) into the equation to solve for a.
-30 = a(0 + 5)(0 - 3)
-30 = a(5)(-3)
-30 = -15a dividing each side by -15 we get
a = 2
The factored form is y = 2(x + 5)(x - 3)
Notice that this is not new. We have already looked at something similar to this.
If the question asked us to write the equation in factored form, we would be finished the question. However, the question is asking us to state the equation in standard form. We must now use FOIL to rearrange the equation.
y = 2(x + 5)(x - 3)
y = 2 (x² - 3x + 5x - 15)
y = 2 (x² + 2x - 15)
y = 2x² + 4x - 30
The equation in standard form is y = 2x² + 4x - 30.
Practice Questions
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