Solve each quadratic equation using the zero-factor property. ...

Question

Solve each quadratic equation using the zero-factor property. See Example 5. 5x² - 3x - 2 = 0

Accepted Answer

Hello, everyone. We are asked to use the zero factor property to solve the quadratic equation for M. The equation we are given is four M squared minus M minus 15 equals zero. Our answer choices are A, the set negative 1/4 3 B, the set negative 3/4 5. See the set 1/4 3 D is set 3/4 5. First, I'm gonna rewrite my equation. So four M squared minus 17 M minus 15 equals zero. Noticing this is already equal to zero means there's one less step we need to do. However, we do need to factor this trinomial into two binomials. So we can have two factors to set equal to zero. So I opened two sets of parentheses and now I'm going to try to factor and figure out what will multiply to make this work. So the first spot of my parentheses, my new binomials has to multiply to four M squared. So sometimes I think two M and two M, I think I'm gonna try four M and M. In this case, my last two terms of my binomials must multiply to a negative and my inner and outer terms will give me my middle term of negative 17 M. So thinking of things that will multiply to a negative 15, I think I'm gonna go with plus three and minus five. So right now I have the parentheses, four M plus three multiplied by the parentheses M minus five. So I'm gonna check to see if this is factored correctly. I do that by multiplying my inner and outer terms. So my inner terms three and M give me three M outer terms negative five and four M give me negative 20 M. And if this adds up to my middle term, it is factored correctly. So it does add up to negative 17 M which means we have factored it correctly. So we can go to the next step. The next step is to set each factor equal to zero on its own. So four M plus three will equal zero and M minus five will equal zero. We're going to solve each of them. So four M plus three equals zero, I subtract three from both sides. So four M equals negative three, divide both sides by four. So we get M equals negative 3/4. And then for M minus five equals zero, I add five to both sides, we get M equals five. So before we say that we are correct, these are our answers, we should check them. So to check them, we plugged them back into the original equation. So M equals negative 3/4 we plug that in as four multiplied by negative 3/4 in parentheses squared minus 17, multiplied by negative 3/4 minus will equal zero. So first, I'm gonna square the negative 3/4. So I have four multiplied by positive 9/ minus 17, multiplied by negative 3/4 minus equals zero. Cross reducing my four and my 16, I get nine fourths and then multiplying 17 and negative 3/ because they're both negative, I will have a positive. So plus, and it'll be plus 51 4th minus 15 equals zero. I'm gonna combine nine fourths and 51 4th which gives me 64th minus 15 equals zero. 64th is the same as 15. So 15 minus 15 equals zero. That one checks. So we know that M equals negative 3/4 is one of our answers. Next, I'm gonna need to check when M equals five. And this one should be a little simpler because it doesn't have all the fractions. So we have four multiplied by five squared minus 17, multiplied by five minus 15 has to equal zero, five. Squared is 25. And I'm gonna do four multiplied by 25 right now. And that's 100 minus 17, multiplied by five is 85 minus equals zero. So 100 minus 85 is 15, 15 minus 15 equals zero. That checks also. So M equals five also checks. So the set that has both negative 3/4 and five is answer choice B have a nice day.

Solve each quadratic equation using the zero-factor property. See Example 5. 5x² - 3x - 2 = 0

Key Concepts

Quadratic Equations

Factoring Quadratic Expressions

Zero-Factor Property

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