Problem 60 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (3-2 x)^{2}=70 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{3 - \sqrt{70}}{2} \) and \( x = \frac{3 + \sqrt{70}}{2} \).

Step 1: Isolate the squared term

The given equation is \( (3-2x)^{2}=70 \). Notice that the squared term \( (3-2x)^{2} \) is already isolated.

Step 2: Apply the square root property

To remove the square, take the square root of both sides of the equation: \[ \sqrt{(3-2x)^2} = \sqrt{70} \]. This simplifies to two equations: \( 3-2x = \sqrt{70} \) and \( 3-2x = -\sqrt{70} \).

Step 3: Solve for x in the first equation

Solve \( 3-2x = \sqrt{70} \): \[ 3 - 2x = \sqrt{70} \] Subtract 3 from both sides: \[ -2x = \sqrt{70} - 3 \] Divide by -2: \[ x = \frac{3 - \sqrt{70}}{2} \]

Step 4: Solve for x in the second equation

Solve \( 3-2x = -\sqrt{70} \): \[ 3 - 2x = -\sqrt{70} \] Subtract 3 from both sides: \[ -2x = -\sqrt{70} - 3 \] Divide by -2: \[ x = \frac{3 + \sqrt{70}}{2} \]

Step 5: Simplify the solutions

The solutions are \( x = \frac{3 - \sqrt{70}}{2} \) and \( x = \frac{3 + \sqrt{70}}{2} \). Both are simplified.

Key Concepts

Solving Quadratic Equations

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \). In the exercise given, we have \( (3-2x)^2=70 \), which can be seen as a form of a quadratic equation. To solve any quadratic equation, there are different methods you can use, like factoring, completing the square, and using the quadratic formula. Here, we use the square root property because the equation is already in a specific squared form. This method is often faster and easier when dealing with perfect squares or expressions raised to the power of two.

Simplifying Radicals

Simplifying radicals helps make equations look simpler and more manageable. A radical expression contains a square root, cube root, or other roots. For instance, \( \sqrt{70} \) is a radical. To simplify radicals, we often look for perfect square factors. However, sometimes, like in our example, the number inside the root is already simplified because 70 is not a perfect square and doesn't have any squared factors besides 1. Thus, \( \sqrt{70} \) is already in its simplest form.

Isolation of Squared Term

Isolating the squared term means getting the \(( expression)^2\) part by itself on one side of the equation. This makes it easier to apply the square root property. In our given problem \((3-2x)^2=70\), the squared term \((3-2x)^2\) is already isolated. If it wasn't, we would first need to use algebraic operations (addition, subtraction, multiplication, or division) to get the squared part by itself. Always look for ways to clear out any coefficients or other terms standing in the way of isolating the squared expression.

Taking Square Roots

Taking the square root of both sides is a crucial step in solving equations like \((3-2x)^2=70\). The square root operation undoes the squaring, simplifying the equation. When you take the square root of both sides, remember that \( \sqrt{a^2} = \pm a \). This is why we get two equations: \(3-2x = \sqrt{70}\) and \(3-2x = -\sqrt{70}\). These have to be solved separately. Solving each one involves straightforward algebra: isolating \(x\) by moving other terms to the other side of the equation and then dividing by the coefficient of \(x\). This is how we get the solutions \( x = \frac{3 - \sqrt{70}}{2} \) and \(x = \frac{3 + \sqrt{70}}{2} \). These are the final simplified solutions.