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

Question

Solve each quadratic equation using the zero-factor property. See Example 5. -4x² + x = -3

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 negative five M squared plus six M equals negative eight. Our answer choices are a negative 4/5 and two B negative five fourths and four C 4/5 and two D five fourths and four. Recall that using the zero factor property, we need to first set our equation equal to zero. So we have negative five M squared plus six M equals negative eight. So I will add eight to both sides. So we have negative five M squared plus six M plus eight equals zero to make this factoring a little easier on myself. I'm gonna factor in negative one out. So I'm gonna do negative one, open parentheses, five M squared minus six M minus eight, closed parentheses equals zero. Now, I want to factor what's in my parentheses that trinomial into two binomials. So I'm opening two sets of parentheses and leaving this equal to zero, negative one is still out front. And I'm trying to think through combinations that will multiply to negative eight and will add to negative six taking into account that I need to multiply one of those values by a five. So thinking about my first term in my new binomials to get five M squared, it's going to have to be five M and M because nothing else is going to multiply to five M squared. Next, knowing that my last term, my trinomial is are negative. I know one of these is positive and one of them is negative. I unfortunately don't really know which one. So I'm gonna say I'll put the minus sign with the five M and the plus sign with the M I can always revise it if it doesn't work how I want it to from there. I'm thinking of things that will multiply to eight and the first one that comes to mind is two and four. So I'm gonna put the four with the five M and the two with the M. So right now I have this factored as negative one multiplied by open parentheses, five M minus four, closed parentheses, multiplied by open parentheses M plus two, closed parentheses equals zero. Now, I need to check to see if this works because it might not recall to check if your factoring is correct, you check the inner and outer solutions. So inner I'd get negative four M outer, I'd get 10 M adding that together. I get a positive six M since I want my middle term to be a negative six M, this means to me that everything is right, except for my signs. So I'm going to rewrite it and change the sign in each binomial. So I'll have negative one multiplied by open parentheses. Five M plus four, closed parentheses, multiplied by the parentheses. M minus two closed parentheses equals zero. And now if I check that, that does work out, so that's the correct factoring from here to solve for M I set each factor with a variable equal to zero. So I'm gonna have five M plus four, equal to zero and M minus two equal to zero in solving those independently, five M plus four equals zero. I subtract four from both sides. So five M would equal negative four divide by five M would equal negative 4/5. And then for M minus two equals zero, I add two to both sides, we get M equals two. Before we say this is our solution, we should check it in our original equation. So starting off checking when M is negative 4/ I have negative five multiplied by negative 4/ squared plus six, multiplied by negative 4/5 would have to equal negative eight. So negative five multiplied by and negative 4/5 squared will be a positive 16, 25th plus six multiplied by negative 4/5 has to equal negative eight. In my first fraction multiplication, I'll cross reduce the five and the 25. So five becomes 1, 25 becomes five. So I have negative 16 5th. When I do that multiplication and six multiplied by negative 4/5 would give me plus a negative 24 5th. The same sequel negative eight for it to be true, negative 16 plus negative 24 is negative 40 divided by five and negative 40 divided by five would be negative eight. So we know that negative 4/5 does work as a solution. And now checking when M equals two, I have negative five multiplied by two squared plus six, multiplied by two S equal negative eight. So negative five multiplied by two squared which is four plus six, multiplied by two has to equal and negative eight. Negative five multiplied by four is negative 20 plus six, multiplied by two is 12, negative 20 plus 12 does equal negative eight. So these are the correct M values M would be negative 4/5 or two. And that's answer choice. A have a nice day.

Solve each quadratic equation using the zero-factor property. See Example 5. -4x² + x = -3

Key Concepts

Quadratic Equations

Zero-Factor Property

Rearranging Equations to Standard Form

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