Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$x^{2}=16$$

Short answer

The solutions of the equation are 4 and -4.

Step-by-step solution

Identify the given quadratic equation

The given equation is \(x^{2} = 16\)

Apply square root to both sides

\(\sqrt{x^{2}} = \sqrt{16}\). Remember, when we take the square root of a square, we always ensure to select both positive and negative values, as both are valid solutions.

Solve for x

This gives us \(x = 4\) and \(x = -4\).

Key concepts

Quadratic Equation

Understanding quadratic equations is crucial to tackling a wide range of problems in algebra. A quadratic equation is any equation that can be rearranged in the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The power of two is what makes this equation 'quadratic'.

To solve quadratic equations, various methods are employed, including factoring, completing the square, using the quadratic formula, and taking square roots. When possible, extracting square roots offers a straightforward solution without needing to manipulate the equation extensively. For the given problem \(x^2=16\), it is easiest to consider the method of taking the square root directly, as the equation is already in a simple form conducive to this approach. We then end up with two possible solutions, as a squared number can have two roots: one positive and one negative.

Square Roots

In the context of solving quadratic equations, square roots are a fundamental concept. A square root of a number is a value that, when multiplied by itself, gives the original number.

The square root symbol is \( \sqrt{} \) and applying it to both sides of an equation is a standard step when the solution involves finding an \( x^2 \) term, like in our example \(x^2 = 16\). It's essential to remember that every positive number has two square roots: a positive and a negative root. This is because \(x^2 = 16\) can be satisfied by both \(4 * 4\) and \( (-4) * (-4) \). Consequently, whenever we apply the square root to solve for \(x^2\), we must consider both \( +\sqrt{x^2}\) and \( -\sqrt{x^2}\) in our solutions.

Radical Expressions

Radical expressions involve roots—a radical symbol denotes taking a root, like the square root. When solving an equation where the solution includes a square root, you might encounter an irrational number. Such solutions are expressed using radical expressions if they can't be simplified to a rational number.

In the exercise \(x^2=16\), extracting the square root gives a whole number. But if the equation were \(x^2=18\), the solutions would be expressed as \(+\sqrt{18}\) and \( -\sqrt{18}\), since \sqrt{18}\ cannot be simplified to an integer. It's crucial to write the solutions as radical expressions in such cases to maintain accuracy. Understanding how to express and simplify radical expressions is an invaluable skill in algebra, ensuring students can accurately convey the result of an equation that does not solve neatly to an integer.