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Solving a Quadratic Equation using Square Roots

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[MUSIC PLAYING] Let's look at our objectives for today. We'll start by reviewing square roots. We'll then look at how to use square roots to solve quadratic equations. And finally, we'll look at how to take the square root of an expression.

Let's start by reviewing square roots. The square root of x is a number whose product with itself is x. For example, 3 times 3 is 9 and negative 3 times negative 3 is 9.

Therefore, the square root of 9 equals 3 and the square root of 9 equals negative 3. There are always two square roots for any positive number. We can use a plus/minus symbol as a shorthand to indicate the positive and negative solutions so we can write the square root of 9 equal to positive and negative 3.

Now, let's talk about solving quadratic equations with square roots. Although factoring is a good method for solving a quadratic equation, when there's no middle term or x term, we can solve the quadratic equation by isolating the variable. Isolating the variable uses inverse operations to undo operations applied to the variable. Inverse operations include addition and subtraction, multiplication and division, and squaring-- taking the square root.

Let's do an example. We want to solve the equation 2x squared minus 8 equals 0. We can start to isolate the x variable by adding 8 to both sides of the equation, which gives us 2x squared equals 8. We then canceled out the 2 being multiplied by the x squared by dividing both sides by 2, which gives us x squared equals 4.

We then take the square root of both sides to undo the 2 exponent that's squaring the x. This gives us x is equal to a plus or a minus 2-- positive and negative 2. Again, taking the square root of a positive number gives two results, one positive and the other negative. So this equation has two solutions.

Now, let's do an example, solving a quadratic equation by taking the square root of an expression. We want to solve 3x plus 3 squared equals 36. Notice that there are multiplication and addition operations being applied to the x variable in the parentheses, which means we undo them last, because, in general, we use the reverse order of operations when solving an equation.

To solve the equation, we start by canceling out the squaring operation by taking the square root of both sides. When we do this, on the left side, we have 3x plus 3 because squaring and square root are inverse operations and they cancel each other out. On the right side, we have a positive or negative 6.

This means that we have two equations and solving each will give us two solutions. To solve the first equation, we start by subtracting 3 on both sides, which gives us 3x equals 3. We then divide by 3 on both sides, which gives us x equals 1 for our first solution.

To solve the second equation, we also begin by subtracting 3 on both sides, this time giving us 3x equals negative 9. We then divide by 3 on both sides to give us x equals negative 3 for our second solution.

Now, we can verify our solution by substituting both solutions separately back into the original equation. Starting with our first solution of x equals 1, substituting it into our equation gives us 3 times 1 plus 3 squared equals 36. We simplify, starting with our parentheses and multiplying 3 times 1, which gives us 3. Adding 3 will give us 6, and 6 squared is 36. So our solution of 1 gives us a true statement, which means it is correct.

Checking our second solution of x equals negative 3, substituting it into the equation. Beginning to simplify with multiplication in the parentheses, 3 times negative 3 gives us negative 9. Plus 3 will give us a negative 6, and negative 6 squared is also 36. So our solution of negative 3 also gives us a true statement, so both solutions are correct.

Let's go over our important points from today. Make sure you get them in your notes so you can refer to them later. The square root of x is a number whose product with itself is x. When solving quadratic equations with no middle or x term, it is easiest to solve by isolating the variable. In taking the square root of a positive number when solving an equation, it may lead to having two solutions to the equation.

So I hope that these important points and examples helped you understand a little bit more about solving quadratic equations using square roots. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.