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Solving a Quadratic Equation by Factoring

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Let's look at our objectives for today. We'll start by looking at the solutions to quadratic equations. We'll then talk about the zero product principle, and finally, we'll do some examples solving quadratic equations by factoring.

Let's start by talking about the solutions to quadratic equations. An equation is a mathematical statement that two quantities have the same value. We use an equal sign between the two quantities to show that they are equal. Quadratic equations can be written in the form ax squared plus bx plus c equals 0, where a, b, and c are real numbers. When quadratic equations are set to 0, the solutions are the values for x that make the expression equal to 0. Therefore, the solutions are commonly referred to as zeroes or roots. And quadratic equations are set equal to 0 in order to solve using different methods, including factoring and the quadratic formula. In our lesson today, we'll focus on solving quadratic equations by factoring.

Now let's look at the zero product principal. Solving quadratic equations by factoring uses the zero product principle. This says that if a product of two factors is zero, then one of the factors must be zero. This is because anything or any value x multiplied by 0 equals 0. So zero can be expressed as the product of zero in any other real number x. The zero product principle can only be used when one side of the quadratic equation is equal to 0.

Let's do an example. We want to solve the quadratic equation x plus 7 times x minus 4 equals 0. The zero product principle tells us that solutions to this equation are when x plus 7 equals 0 and when x minus 4 equals 0. So we can solve each of these equations to find our two solutions. Solving the first equation, we start by subtracting 7 on both sides, which gives us x equals negative 7 for our first solution. For the second equation, we add 4 to both sides, which gives us x equals a positive 4. This means that the quadratic equation has two solutions-- x equals negative 7 and x equals 4.

Now let's do an example, solving a quadratic equation by factoring. We want to solve the equation x squared plus 7x plus 10 equals 0. If we can factor this quadratic expression on the left side of the equation, we can use the zero product principle to find solutions. Therefore, we want to find two numbers that multiply to 10 and add to 7.

We start by looking at the numbers which multiply to 10. The factors of 10 are 1 and 10, negative 1 and negative 10, 2 and 5, and negative 2 and negative 5. The pair of these numbers which also add to 7 are 2 and 5. So we can rewrite our expression in factored form as x plus 2 times x plus 5, which will be equal to 0. We then can set each factor equal to 0 to write two separate equations, so we have x plus 2 equals 0 and x plus 5 equals 0. Solving the first equation by subtracting 2 on both sides gives us x equals negative 2, and solving the second equation by subtracting 5 from both sides will give us x equals negative 5. So our two solutions are x equals negative 2 and x equals negative 5.

We can verify each solution by substituting it back into the original equation and simplifying. Substituting our first solution of negative 2 into the equation gives us negative 2 squared plus 7 times negative 2 plus 10 equals 0. Simplifying the exponent gives us 4. Multiplying 7 times negative 2 gives us a negative 14 plus 10 equals 0. Adding and subtracting from left to right gives us 0 equals 0, which is a true statement, so x equals negative 2 is correct.

Substituting negative 5, our second solution, into the equation gives us negative 5 squared plus 7 times negative 5 plus 10 equals 0. Simplifying our exponent gives us 25. 7 times negative 5 is negative 35 plus 10 equals 0. And adding and subtracting from left to right gives us 0 equals 0, so our second solution of negative 5 is also correct.

Here's our next example. We want to solve x squared plus 5x minus 24 equals 12. We know that in order to use the zero product principle, we need to set the equation equal to 0. To do this, we can subtract 12 from both sides, which will give us x squared plus 5x minus 36 equals 0. So now we can see if we can factor the quadratic expression to use the zero product principle to find the solutions.

We need to find two numbers that multiply to negative 36 and add to 5. We start by looking at the numbers that multiply to negative 36. The factor pairs of negative 36 are listed here. The pair of numbers that multiply to negative 36 and also add to 5 are negative 4 and positive 9. So we can factor our expression as x minus 4 times x plus 9, which will be equal to 0.

When x minus 4 is equal to 0, x equals 4, and when x plus 9 is equal to 0, x equals negative 9, so our two solutions are x equals 4 and x equals negative 9. Here's our last example. We want to solve the equation 4x squared plus 16x equals 0. We notice that each term has common factors of 4 and x which can be factored out. Therefore, our equation becomes 4x times x plus 4, which is equal to 0.

We can then use the zero product principle to set each factor equal to 0 to find the solutions. So 4x equals to 0-- we solve by dividing by 4 on both sides, which gives us x equals 0. And x plus 4 equals 0, we solve by subtracting 4 on both sides, which gives us x equals negative 4. So the two solutions to this equation are x equals 0 and x equals negative 4.

Let's go over our important points from today. Make sure you get them in your notes so you can refer to them later. When quadratic equations are set equal to 0, the solutions are the values for x that make the expression equal to 0. Therefore, the solutions are commonly referred to as zeroes or roots. Quadratic equations are set equal to 0 in order to solve using different methods, including factoring and the quadratic formula. And the zero product principle says that if a product of two factors is equal to 0, then one of the factors must be 0.

So I hope that these important points and examples helps you understand a little bit more about solving a quadratic equation by factoring. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

00:00 – 00:35 Introduction

00:36 – 01:21 Solutions to Quadratic Equations

01:22 – 02:42 Zero Product Principle

02:43 – 07:13 Examples Solving Quadratic Equations by Factoring

07:14 – 08:05 Important to Remember (Recap)