Question

Solve the equation. $$6(x-14)^{2}=0$$

Short answer

The solution to the equation is \(x=14\).

Step-by-step solution

Cancel out the constant term

First, get rid of the constant term 6 by dividing both sides of the equation by 6. With both sides divided by 6, the equation becomes \((x-14)^{2}=0\).

Take the square root

The next step is to take the square root of both sides to remove the power of 2. Doing so gives \(x-14=0\).

Solve for x

Finally, solve for x by adding 14 to both sides of the equation. This leads to the solution \(x=14\).

Key concepts

Quadratic Equations

Quadratic equations are algebraic expressions that are set equal to zero and typically take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents a variable. These types of equations are called 'quadratic' because \(quad\) refers to the square term \(x^2\).

Quadratic equations can have either one or two real solutions, or even complex solutions depending on the discriminant value which is calculated as \(b^2 - 4ac\). Solutions to a quadratic equation are found using methods such as factoring, completing the square, or applying the quadratic formula. In the provided exercise \(6(x-14)^{2}=0\), the quadratic expression is already in the form of a perfect square, which simplifies the solving process considerably.

Square Roots

The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). Mathematically, the square root of \(x\) is represented as \(\sqrt{x}\). In context to quadratic equations, taking the square root is a crucial step in solving when the equation is a perfect square, meaning the quadratic is in the form of \(a(x-h)^2=k\), where \(h\) and \(k\) are constants.

When we take the square root of both sides, as seen in the provided exercise \(\sqrt{(x-14)^{2}}=\sqrt{0}\), we are essentially undoing the square of the binomial, which brings us one step closer to identifying the value of \(x\). However, one should always remember that the square root operation can yield both positive and negative results \(\pm\) since \(\sqrt{x^2}=\pm x\). Therefore, we need to consider both possible values when solving quadratic equations.

Equation Operations

Solving equations often involves carrying out a series of operations to isolate the variable. These operations include addition, subtraction, multiplication, and division, which are used to maintain the balance of the equation (what you do to one side of the equation must be done to the other side).

In the case of the given exercise \(6(x-14)^{2}=0\), dividing both sides by 6 simplifies the equation without affecting the balance, leading us to \(x-14)^{2}=0\). Then, we proceed with the square root operation to eliminate the square, setting the stage for simple addition or subtraction to solve for \(x\). It's important to perform these operations step by step, ensuring that each action keeps the equation in balance and moves you closer to finding the solution for the variable.