Solve each equation. (x-2)(x^2-8 x+7)=0

Name: Problem 83, Chapter 6: Elementary and Intermediate Algebra
Uploaded: 2020-04-15T18:23:14-08:00
Duration: 6 min 27 s
Channel: Karuna Ramroop
Description: Solve each equation. (x-2)(x^2-8 x+7)=0

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental concept in algebra stating that if the product of two or more factors equals zero, then at least one of the factors must be zero. This property allows us to break down complex equations into simpler ones by setting each factor equal to zero and solving for the variable.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors, which are simpler polynomials. This process simplifies the task of solving equations, as factorized polynomials are easier to work with, allowing us to apply the Zero Product Property to find the roots.

Solving Quadratic Equations

Solving quadratic equations is a key skill in algebra. When a quadratic expression can be factored into a product of two binomials, it becomes straightforward to set each factor equal to zero and solve for the variable. This technique simplifies finding the roots of quadratic equations without resorting to the quadratic formula.

Transcript

00:01 Alright, so we need to solve this equation for x.

00:05 So looking at this, i can see that it is partially factored on the left side, and the right side is equal to zero.

00:14 So i say partially because we can see that we have one factor here, x minus 2, and that is fully factored, and there's nothing you can do to it.

00:22 But it's being multiplied by another factor that can be factored down a little bit more.

00:31 So what i want to do is i want to finish factoring this second factor here so that we can use the zero factor property to help us out.

00:41 So on the next page, we are going to factor out x squared minus 8x plus 7.

00:51 So to factor this, we're going to use the ac method.

00:55 We can see that our a is an invisible 1 in front of x squared, our b is negative 8, and our c is 7.

01:01 So a times c in this case is going to be 1 times 7, which is just 7.

01:07 And our b is negative 8.

01:11 So we need to find two numbers that multiply to positive 7, and those same two numbers need to add up to negative 8.

01:18 So the only factors of 7 are 1 and 7.

01:22 So that means we need to figure out what signs to put on our 1 and 7 to make both of these equations true.

01:31 So if they multiply to 1, to positive 7, either both the one in the 7 are positive or both of them are negative.

01:42 Looking at the second equation, if they add up to negative 8, that means they both have to be negative.

01:50 So we're going to make both of them negative, and we're going to make sure they both satisfy these two equations.

01:56 So negative 1 times negative 7 is positive 7, so this is true, and negative 1 plus negative 7 is negative 8.

02:04 So that is also true.

02:05 So we found our two numbers, they are negative 1 and negative 7.

02:11 Now we're going to split that middle term into these two numbers.

02:16 So we're going to first, we're just going to split it into negative 1x plus negative 7x.

02:28 Then i'm going to bring down the other terms.

02:33 So now i'm going to group the first two terms and group the last two terms.

02:38 And we're going to see what we can factor out of each grouping.

02:42 So looking at the first grouping, x squared minus 1x, the only thing i can factor out here is an x.

02:48 So that's what we're going to do.

02:50 I'm going to factor out an x, and then we get x minus 1...

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