Solve each quadratic equation by the square root property. If possible, simplify radicals or rat

Solve each quadratic equation by the square root property....

Key Concepts

Quadratic Equation

A quadratic equation is any equation in which the highest power of the unknown variable is 2. It typically takes the form ax² + bx + c = 0, where a, b, and c are constants. Understanding quadratic equations is key to many areas of algebra and calculus, and they can be solved using various methods such as factoring, completing the square, and the quadratic formula.

Square Root Property

The square root property is a method for solving quadratic equations that have been simplified to the form x² = k, where k is a constant. This property states that if x² = k, then x can be found by taking the positive and negative square roots of k, resulting in x = ±?k. This method is particularly effective when the equation is easily manipulated into this form.

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression so that there are no perfect square factors (other than 1) inside the radical sign. This process often includes extracting factors that are perfect squares and rationalizing denominators in order to present the expression in its simplest form. It is an important step to ensure precision and clarity in the final answer when working with square roots.

Transcript

00:01 For this problem, we've been asked to solve for x using the square root property.

00:06 The square root property states that for some variable squared equal to a real number, the value of that variable is equal to the positive and negative square root of the real number.

00:19 In this equation, our squared variable is not alone on one side of the equation, so the square root property cannot be immediately used.

00:30 Another way of thinking about this is solving the equation for x using inverse operations, much like you would in any other circumstance.

00:40 And we'll talk about how the square root property demonstrates that in just a moment.

00:45 If i want to start solving this equation for x, i need to get it by itself.

00:51 So the first task would be to get the term with x by itself, which is 3x squared.

00:57 To do that, i would add two to each side.

01:04 That will leave 3x squared, which is the term with the variable by itself on the left, equal to positive 2.

01:15 Now that i have the term with the variable by itself, i want to remove its coefficient 3.

01:21 Since 3 is being multiplied with x squared, i would divide by 3 to get rid of it.

01:28 If i divide by 3 on the left, i must do so on the right to keep the equation balanced.

01:35 On the left, that will leave x squared by itself equal to positive 2 over 3 on the right.

01:47 The last step to isolate x would be to remove the exponent 2.

01:52 This is where the square root property comes in.

01:55 The square root property states that the square root, or taking a square root, an inverse operation that will remove the exponent 2.

02:06 A square root undoes a square.

02:09 So that's what we'll do here.

02:11 To remove this exponent 2, we can take a square root...

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