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

Question

Solve each quadratic equation using the zero-factor property. See Example 5. 25x² + 30x + 9 = 0

Accepted Answer

Welcome back everyone. In this problem, we want to use the zero factor property to solve the quadratic equation for M. And that quadratic equation is 49 M squared plus 28 M plus four equals zero. For our answer choices A is 2/7 B is negative 2/7 C is negative 2/7 and 2/7 and D is negative seven halves and positive seven halves. Now, let's see if we can figure it out. What do we already know? Well, we know that we want to use the zero factor property. What does that mean? Well, recall that the zero factor property say says that if A and B are real numbers, OK, then once a multiplied by B equals zero, either A equals zero B equal zero, R both equal zero. So how this helps us is that with our quadratic equation, if we can write it in the form of a product uh in the form of two linear expressions multiplying each other, then we'll be able to factorize and apply that, that, that property to help us solve. So let's see if we can do that. So we have 49 M squared plus 28 M plus four equals zero. Now, does this quadratic equation look familiar? Doesn't it look like something you've seen before? Don't the numbers at either end of the expression look pretty interesting. That's because this is a perfect square and recall that for a perfect square. What that means is that if we have a quadratic equation in the form A squared plus two multiplied by A B plus B squared, then we can write it as a perfect square for A plus BR squared. And that's what's happening in this case because notice that for 49 M squared, what this really is saying this is really seven M all squared. So 49 M squared is a square number. Likewise four is a square number because four could be written as two squared and 28 M is the product of two multiplied by those roots. That is two multiplied by seven M multiplied by two. So because of that, we can rewrite our expression in that form and then make our perfect square. So this is telling us that we can rewrite it as seven M all squared plus two multiplied by seven M multiplied by two plus two squared equals zero where A is seven M and B is two. So applying the property, this means we can rewrite it as seven M plus two, all squared equals zero. No, that's the case. That means we can simplify further if we want to get seven M plus two, we can square root both sides. So that means the spirit of seven M plus two, all squared is seven M plus two. And the spirit of zero, of course, that would be plus or minus zero but zero case neither positive nor negative. So that would just be equal to zero. Now, if seven M plus two equals zero, that means seven M equals negative two and M would equal negative two divided by seven or negative 2/7. We just want to make sure we are right before we continue. So let's try to put it back into our equation. Let's test it. So testing it in our equation that would now become 49 multiplied by negative 2/7 squared plus 28 multiplied by negative 2/ plus four equals zero. OK? And we're checking if it really equals zero. That's how we know that we're right if it works in our equation and now when we, when we do our calculations negative 2/7 squared, that's negative 2/7 multiplied by negative 2/7. So that's 4, 49th, 28 multiplied by negative 2/7. Well, seven can factor into 28 and it can factor out four. So I can rewrite that as four multiplied by negative two. And I add fourth to that no, 49th factors are 49th in our first term. So that leaves us with 44 multiplied by negative two. That's negative eight and four minus eight gives us negative four, negative four plus four gives us zero. So what's on our left hand side works out to be on our right hand side that tells us that the equation is true. Four me equals negative to seventh. Therefore, our answer has to be b thanks a lot for watching everyone. I hope this video helped.

Solve each quadratic equation using the zero-factor property. See Example 5. 25x² + 30x + 9 = 0

Key Concepts

Quadratic Equations

Factoring Quadratic Expressions

Zero-Factor Property

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