Problem 50 - Algebra 1: Concepts and Skills

Question

Solve the equation. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}+81=144 $$

Step-by-Step Solution

Answer

The solution of the equation is \(x = ± √63\).

Step 1: Rewrite The Equation

The first step required is to rewrite the equation in a standard quadratic form, which is \(Ax^2 + Bx + C = 0\). This is done by subtracting 81 from both sides of the equation, which gives us \(x^{2} - 144 + 81 = 0\), further simplification leads to \(x^{2} - 63 = 0\).

Step 2: Solve For 'x'

Solve for 'x' by moving 63 to the other side, this results in \(x^{2} = 63\). To solve for 'x', take the square root of both sides, resulting in \(x = ± √63\).

Step 3: Simplify 'x' If Possible

Checking if the square root of 63 can be simplified, it is found that it cannot be further simplified. Therefore, the solutions remain as \(x = ± √63\).

Key Concepts

Radical Expressions

Understanding radical expressions is essential when solving equations that involve roots. A radical expression is an expression that includes a square root, cube root, or higher roots. For example, in the exercise \( x^2 + 81 = 144 \), after simplifying the equation to \( x^2 = 63 \), the next step is to take the square root of both sides. Taking the square root is the inverse operation of squaring and is represented by the radical symbol \sqrt{\cdot}.
When we write \( x = \pm \sqrt{63} \) as the solution, we are expressing x as a radical expression. It's important to check if the radical can be simplified by finding perfect square factors. In this case, 63 does not have a perfect square factor, so the radical expression remains as it is. A vital point to remember is that whenever you take the square root of a number in an equation, you consider both the positive and negative roots, hence the \pm symbol is used.

Quadratic Formula

The quadratic formula is a reliable method for solving any quadratic equation of the form \(Ax^2 + Bx + C = 0\). In the given exercise, although it is not necessary to use the quadratic formula because the equation simplifies neatly, it's invaluable to know this formula for more complex equations. The quadratic formula is \( x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \).

• It takes the coefficients A, B, and C from the standard form of a quadratic equation.
• The symbol \pm indicates that there will be two solutions: one positive root and one negative root.
• The expression \(B^2 - 4AC\) under the square root is known as the discriminant, which determines the nature of the roots.

Using the quadratic formula can help you find solutions that are not immediately apparent and provides a systematic approach that works even when the equation cannot be easily factored or simplified.

Square Roots

Square roots are fundamental when solving quadratic equations, as seen in the step where \( x^2 = 63 \) leads to \( x = \pm \sqrt{63} \). The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\).
Understanding square roots is crucial because:

• It helps determine the solutions to quadratic equations where the variable is squared.
• It helps us simplify radical expressions when possible by finding and extracting square factors.
• Many mathematical problems in various fields rely on the concept of taking square roots.

For the exercise \( x^{2} = 63 \), the square root of 63 does not simplify to an integer, but recognizing when and how to take the square root is key to solving these types of problems.