| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| {{ 'ml-lesson-time-estimation' | message }} |
Here are a few recommended readings before getting started with this lesson.
The following applet shows a quadratic equation whose left-hand side is written in factored form. Using the Zero Product Property find the solutions of the equation. Round to the nearest tenth if necessary.
Dolphins jump out of the water to improve their navigation and to see the surface of the ocean. They also do it for fun.
The following equation models the height h in meters of a jumping dolphin t seconds after it leaves the water.The dolphin is in the water when h=0.
Rewrite -5t2 as t(-5t)
Commutative Property of Multiplication
Factor out t
seeits factors. Consider the following expression.
It is known that a=4 and c=3, so ac=12>0. Therefore, the factors must have the same sign. Also, b=13. Since the sum of the factors is positive and they must have the same sign, both factors must be positive. All positive factor pairs of 12 can now be listed and their sums checked.
Factors of ac | Sum of Factors |
---|---|
1 and 12 | 1+12=13 ✓ |
2 and 6 | 2+6=8 × |
3 and 4 | 3+4=7 × |
In this case, the correct factor pair is 1 and 12. The following table sums up how to determine the signs of the factors based on the values of ac and b.
ac | b | Factors |
---|---|---|
Positive | Positive | Both positive |
Positive | Negative | Both negative |
Negative | Positive | One positive and one negative. The absolute value of the positive factor is greater. |
Negative | Negative | One positive and one negative. The absolute value of the negative factor is greater. |
Such analysis makes the list of possible factor pairs shorter.
The above method can be used to factor any quadratic trinomial. It is particularly useful for trinomials in the forms ax2+bx or ax2+c. However, in these two cases there are other approaches that might be used as well.
Please note that not every quadratic trinomial can be factored. When a pair of integers whose product is equal to a⋅c and whose sum is b cannot be found, the trinomial cannot be factored using the described method.
x2−x+1, a⋅c=1, b=-1 | |
---|---|
Product of Factors | Sum of Factors |
1⋅1=1 | 1+1=2 × |
-1(-1)=1 | -1+(-1)=-2 × |
Zosia is a student studying how many Dolphins swim past a certain area of cove. She marks off a rectangle using rope where she will count how many dolphins pass through in any given day. The study area's rectangle is equal to 42 square meters. However, the dimensions of the rectangle are unknown.
What are the width and the length of the rectangle?Based on the given information, write and solve a quadratic equation for x.
Distribute (4x−7)
Distribute 2x & -1
Subtract term
LHS−42=RHS−42
Pair of Factors | Sum of Factors |
---|---|
1 and -280 | 1+(-280)=-279 × |
2 and -140 | 2+(-140)=-138 × |
4 and -70 | 4+(-70)=-66 × |
5 and -56 | 5+(-56)=-51 × |
7 and -40 | 7+(-40)=-33 × |
8 and -35 | 8+(-35)=-27 × |
10 and -28 | 10+(-28)=-18 ✓ |
14 and -20 | 14+(-20)=-6 × |
Split into factors
Factor out 2x
Factor out -7
Use the Zero Product Property
(I): LHS−5=RHS−5
(I): LHS/4=RHS/4
(I): Put minus sign in front of fraction
(I): Calculate quotient
2x−1 | 4x−7 | |
---|---|---|
x=-1.25 | 2(-1.25)−1=-3.5 × | 4(-1.25)−7=-12 × |
x=3.5 | 2(3.5)−1=6 ✓ | 4(3.5)−7=7 ✓ |
It can be seen that x=-1.25 results in a negative width and length, which is not logical for the dimensions of a rectangle. Therefore, the only valid solution is x=3.5. The width of the rectangle is 6 meters and the length is 7 meters.
Rewrite the equation in standard form. Factor the left-hand side of the rewritten equation.
Pair of Factors | Sum of Factors |
---|---|
-1 and -225 | -1+(-225)=-226 × |
-3 and -75 | -3+(-75)=-78 × |
-5 and -45 | -5+(-45)=-50 × |
-9 and -25 | -9+(-25)=-34 × |
-15 and -15 | -15+(-15)=-30 ✓ |