Problem 20 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}=17 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}=17\) are \(x = ±\sqrt{17}\).

Step 1: Identify the type of equation

The given equation \(x^{2}=17\) is a quadratic equation, which has the general form \(ax^{2} + bx + c = 0\). In this case, \(a = 1\), \(b = 0\), and \(c = -17\).

Step 2: Apply the square root method

To find a solution for \(x\), the square root method can be applied to both sides of the equation, which essentially involves taking the square root of both sides. Considering both the positive and negative roots, the solution may be written as: \(x = ±\sqrt{17}\).

Step 3: Simplify the solution

The value inside the square root, 17, can't be simplified further as it's a prime number, so the solutions to this equation remain as \(x = ±\sqrt{17}\).

Key Concepts

The Square Root Method

The square root method is a straightforward approach for solving quadratic equations that take the form of \(x^{2} = a\), where \(a\) is any non-negative number. The method proceeds by taking the square root of both sides of the equation to solve for \(x\). It's essential to include both positive and negative solutions since both \( (\text{positive number})^2 \) and \( (\text{negative number})^2 \) will result in the same value of \(a\).For example, with the equation \(x^{2} = 17\), applying this method involves two steps:

• Take the square root of both sides: \(\sqrt{x^{2}} = \pm \sqrt{17}\).
• Since \(\sqrt{x^{2}}\) is \(x\), the solutions are \(x = \pm \sqrt{17}\).

Keep in mind that this method is most effective for equations that do not include a linear term (\(bx\)) and have been arranged so that one side of the equation is zero.

Understanding Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher. The square root symbol \(\sqrt{\phantom{x}}\) specifically denotes a square root, which answers the question: 'What number, when multiplied by itself, will equal the given value?' The expression under the square root sign is called the radicand.When simplifying radical expressions, the goal is to find the simplest form. A radical is in its simplest form when:

• The radicand has no perfect square factors except 1.
• The radicand is not a fraction.
• No radical appears in the denominator of a fraction.

In our exercise, \(x = \pm \sqrt{17}\), there are no perfect squares that can be factored from 17 since it's a prime number. Hence, \(\sqrt{17}\) is already in its simplest form and these are the solutions to the equation in radical form.

Quadratic Equation Properties

Quadratic equations typically have the standard form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Crucial properties of quadratic equations include:

• They have at most two real solutions. These solutions can be integers, fractions, irrational numbers, or even complex numbers if we allow for non-real solutions.
• The sign of the leading coefficient \(a\) determines whether the parabola opens upward (\(a > 0\)) or downward (\(a < 0\)).
• The solutions of the equation can also be found by factoring, completing the square, or using the quadratic formula when the square root method isn't applicable.

In the exercise \(x^{2} = 17\), there are no \(b\) or \(c\) terms, and the coefficient \(a\) is 1, implying an upward-opening parabola. This specific form makes it ideal for the square root method and demonstrates the importance of understanding the structure of a quadratic equation to choose the appropriate solving technique.