Problem 30 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ (4 x-1)^{2}=16 $$

Step-by-Step Solution

Answer

The solutions to the problem are \(x = 1.25\) and \(x = -0.75\) .

Step 1: Isolate the square

First, isolate the squared term. The equation should look like this: $(4x - 1)^2 = 16$ .

Step 2: Take the square root of both sides

Take the square root of both sides of the equation. Remember to account for the positive and negative roots of the second side.This looks like this: \[±\sqrt {(4x - 1)^2} = ±\sqrt {16}\]Reducing both sides gives us:\[± (4x-1) = ±4\] .

Step 3: Split into two equations

Since we have ± signs, split the equation into two different equations to solve for x.These are: \[4x - 1 = 4\] and \[4x - 1 = -4\] .

Step 4: Solve the two equations

Finally, solve each equation separately. For \[4x - 1 = 4\] , add 1 to both sides to get \[4x = 5\] and then divide both sides by 4 to get \(x = \frac{1.25}{1}\) . For \[4x - 1 = -4\] , add 1 to both sides to get \[4x = -3\] and then divide both sides by 4 to get \(x = \frac{-3}{4} \) .

Key Concepts

Solving Quadratic Equations

Quadratic equations are a central topic in algebra that involve variables raised to the second power (x^2). They generally take the form ax^2 + bx + c = 0. Solving these equations means finding the value(s) of x that make the equation true. There are several methods for solving quadratics, such as factoring, completing the square, using the quadratic formula, and employing the square root property as illustrated in the example provided.

When approaching a quadratic equation, the goal is to determine the 'roots' or 'solutions', which are the values that satisfy the equation. These roots can be real or complex numbers, and a single quadratic equation can have up to two distinct solutions. Understanding the square root property is essential for solving quadratics when they can be manipulated into a form that isolates a squared term.

Isolating the Squared Term

Before employing the square root property, it's crucial to isolate the squared term on one side of the equation. This involves moving all other terms to the opposite side through basic algebraic operations such as addition, subtraction, multiplication, and division.

In the given exercise, the squared term is (4x - 1)^2, and it is already isolated on one side of the equation, equal to 16. This step simplifies the equation and sets the stage for the application of the square root property. The process of isolating the term makes it possible to deal directly with the exponent and prepare for extracting the roots. In some cases, further simplification may be necessary to reach a form where the squared term stands alone before proceeding to the next step.

Taking Square Roots

Taking the square root of both sides of an equation is a valid operation provided that you account for both the positive and negative square roots. This stems from the fact that squaring any real number, whether positive or negative, yields a positive result.

In the step-by-step solution, after the squared term is isolated, we take the square root of both sides which gives us ±√(4x - 1)^2 = ±4. The '±' symbol indicates that both the positive and negative roots must be considered because (+4)^2 and (-4)^2 both yield 16. This step is critical and is a common point where mistakes can be made if one forgets to include the negative solution. Including both positive and negative solutions ensures that all possible solutions to the equation are taken into account.

Equation Solutions

The final stage in solving a quadratic equation using the square root property is to find the specific solutions for the variable. After taking the square root, the resultant expressions can yield either one equation or two separate equations if there is a '±' symbol present. In the example provided, we get two separate linear equations to solve for x.

These equations are 4x - 1 = 4 and 4x - 1 = -4. By solving these equations, we find the two possible values of x that satisfy the original quadratic equation. It is essential to treat each resultant equation independently and solve for x by performing appropriate algebraic operations. By isolating x, we get x = 5/4 and x = -3/4, which are the two solutions to the original equation. Remember that the number of solutions to a quadratic equation can be zero (no real solutions), one (if both solutions are the same, often called a 'repeated root'), or two distinct real solutions.