solve each quadratic equation by factoring and applying the zero product property. 12 w^2-41 w+2

Solve each quadratic equation by factoring and applying the...

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation generally in the form ax² + bx + c = 0, where a, b, and c are constants with a ? 0. Solving quadratic equations involves finding the values of x that satisfy the equation, and one common method is factoring when the quadratic is factorable.

Factoring

Factoring is the process of rewriting a polynomial as a product of two or more simpler expressions (factors). In the context of quadratic equations, factoring allows the quadratic to be expressed as the product of two binomials. This simplification is key to applying the zero product property and finding the solutions of the equation.

Zero Product Property

The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. After factoring a quadratic equation into factors, setting each factor equal to zero provides a direct way to find the roots or solutions of the equation.

Transcript

00:01 In this problem, we're being asked to solve the given equation by factoring.

00:03 Well, first off, is said equal to zero, so that's good.

00:06 So the factor, we should try and factor out of the greatest common factor if there is one, but in this case, there isn't.

00:11 So we have a problem where our trinomial has a leading coefficient greater than one.

00:16 There's multiple ways to factor this.

00:18 I end up doing this by using a grouping method.

00:20 So i'll begin by multiplying our first and last coefficients, while 12 times 24 is equal to 288.

00:27 So now we have to find two integers that multiply.

00:30 Apply to 288 that will add to our middle coefficient of negative 41.

00:35 So in this case, i'm thinking negative 32 and negative 9, because negative 32 plus negative 9 is negative 41, and negative 32 times negative 9 is 288.

00:47 Perfect.

00:48 So next, i'm going to rewrite our problem, but in place of negative 41w, i'm going to substitute in negative 32w minus 9w.

00:57 And then we'll brisk bring down our first and last terms.

01:01 So now, as you can see, we have four terms, so now we can factor by group it.

01:06 So we'll split our problem in half and factor out the greatest common factor from the first two terms, which is going to be 4w.

01:14 So then when we divide both terms by 4w, that leaves us with 3w minus 8...

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