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Solve a Quadratic Equation by Factoring

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❶Be careful not to accept this as the solution, but switch signs so the larger product agrees in sign with the middle term. We must find numbers that multiply to give 24 and at the same time add to give -

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FACTORING BY GROUPING

When the products of the outside terms and inside terms give like terms, they can be combined and the solution is a trinomial.

This method of multiplying two binomials is sometimes called the FOIL method. It is a shortcut method for multiplying two binomials and its usefulness will be seen when we factor trinomials. Again, maybe memorizing the word FOIL will help. Not only should this pattern be memorized, but the student should also learn to go from problem to answer without any written steps.

This mental process of multiplying is necessary if proficiency in factoring is to be attained. As you work the following exercises, attempt to arrive at a correct answer without writing anything except the answer. The more you practice this process, the better you will be at factoring. Now that we have established the pattern of multiplying two binomials, we are ready to factor trinomials.

We will first look at factoring only those trinomials with a first term coefficient of 1. Since this is a trinomial and has no common factor we will use the multiplication pattern to factor. We will actually be working in reverse the process developed in the last exercise set. We now wish to fill in the terms so that the pattern will give the original trinomial when we multiply.

Remember, the product of the first two terms of the binomials gives the first term of the trinomial. We must now find numbers that multiply to give 24 and at the same time add to give the middle term.

Notice that in each of the following we will have the correct first and last term. Some number facts from arithmetic might be helpful here. The product of two odd numbers is odd. The product of two even numbers is even.

The product of an odd and an even number is even. The sum of two odd numbers is even. The sum of two even numbers is even.

The sum of an odd and even number is odd. Thus, only an odd and an even number will work. We need not even try combinations like 6 and 4 or 2 and 12, and so on. Here the problem is only slightly different. We must find numbers that multiply to give 24 and at the same time add to give - You should always keep the pattern in mind. The last term is obtained strictly by multiplying, but the middle term comes finally from a sum.

Knowing that the product of two negative numbers is positive, but the sum of two negative numbers is negative, we obtain. We are here faced with a negative number for the third term, and this makes the task slightly more difficult. Since can only be the product of a positive number and a negative number, and since the middle term must come from the sum of these numbers, we must think in terms of a difference.

We must find numbers whose product is 24 and that differ by 5. Furthermore, the larger number must be negative, because when we add a positive and negative number the answer will have the sign of the larger.

Keeping all of this in mind, we obtain. The order of factors is insignificant. The following points will help as you factor trinomials: When the sign of the third term is positive, both signs in the factors must be alike-and they must be like the sign of the middle term. When the sign of the last term is negative, the signs in the factors must be unlike-and the sign of the larger must be like the sign of the middle term.

In the previous exercise the coefficient of each of the first terms was 1. When the coefficient of the first term is not 1, the problem of factoring is much more complicated because the number of possibilities is greatly increased.

Having done the previous exercise set, you are now ready to try some more challenging trinomials. Notice that there are twelve ways to obtain the first and last terms, but only one has 17x as a middle term.

You could, of course, try each of these mentally instead of writing them out. In this example one out of twelve possibilities is correct.

Thus trial and error can be very time-consuming. Even though the method used is one of guessing, it should be "educated guessing" in which we apply all of our knowledge about numbers and exercise a great deal of mental arithmetic. In the preceding example we would immediately dismiss many of the combinations. Since we are searching for 17x as a middle term, we would not attempt those possibilities that multiply 6 by 6, or 3 by 12, or 6 by 12, and so on, as those products will be larger than Also, since 17 is odd, we know it is the sum of an even number and an odd number.

All of these things help reduce the number of possibilities to try. First find numbers that give the correct first and last terms of the trinomial. Then add the outer and inner product to check for the proper middle term. First we should analyze the problem. The last term is positive, so two like signs. The middle term is negative, so both signs will be negative. The factors of 6x2 are x, 2x, 3x, 6x. The factors of 15 are 1, 3, 5, Eliminate as too large the product of 15 with 2x, 3x, or 6x.

Try some reasonable combinations. These would automatically give too large a middle term. See how the number of possibilities is cut down. The last term is negative, so unlike signs.

We must find products that differ by 5 with the larger number negative. We eliminate a product of 4x and 6 as probably too large. Remember, mentally try the various possible combinations that are reasonable. This is the process of "trial and error" factoring. You will become more skilled at this process through practice.

Be careful not to accept this as the solution, but switch signs so the larger product agrees in sign with the middle term. By the time you finish the following exercise set you should feel much more comfortable about factoring a trinomial.

Identify and factor the differences of two perfect squares. Identify and factor a perfect square trinomial. In this section we wish to examine some special cases of factoring that occur often in problems.

If these special cases are recognized, the factoring is then greatly simplified. Recall that in multiplying two binomials by the pattern, the middle term comes from the sum of two products. From our experience with numbers we know that the sum of two numbers is zero only if the two numbers are negatives of each other. When the sum of two numbers is zero, one of the numbers is said to be the additive inverse of the other.

In each example the middle term is zero. This is the form you will find most helpful in factoring. Special cases do make factoring easier, but be certain to recognize that a special case is just that-very special. In this case both terms must be perfect squares and the sign must be negative, hence "the difference of two perfect squares. The sum of two squares is not factorable. You must also be careful to recognize perfect squares.

Remember that perfect square numbers are numbers that have square roots that are integers. Also, perfect square exponents are even. Students often overlook the fact that 1 is a perfect square. Thus, an expression such as x 2 - 1 is the difference of two perfect squares and can be factored by this method. Another special case in factoring is the perfect square trinomial. Observe that squaring a binomial gives rise to this case. We recognize this case by noting the special features.

Three things are evident. The first term is a perfect square. The third term is a perfect square. The middle term is twice the product of the square root of the first and third terms. For factoring purposes it is more helpful to write the statement as. To factor a perfect square trinomial form a binomial with the square root of the first term, the square root of the last term, and the sign of the middle term, and indicate the square of this binomial.

Always square the binomial as a check to make sure the middle term is correct. Find the key number of a trinomial. Use the key number to factor a trinomial. In this section we wish to discuss some shortcuts to trial and error factoring. These are optional for two reasons. First, some might prefer to skip these techniques and simply use the trial and error method; second, these shortcuts are not always practical for large numbers.

However, they will increase speed and accuracy for those who master them. The first step in these shortcuts is finding the key number. Browse Articles By Category Browse an area of study or degree level. Are High Schools Failing Students?

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Factoring. The process of factoring is essential to the simplification of many algebraic expressions and is a useful tool in solving higher degree equations. In fact, the process of factoring is so important that very little of algebra beyond this point can be accomplished without understanding it. Topics from help homework you'll be able to complete: Finding the prime homework of a help Finding the least common multiples using prime factorizations Simplifying fraction notation and finding equivalent expressions Factoring out variables Factoring out combined numbers and variables Using division to factor problems Factoring by business.

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