Problem 48 - 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. $$ 20-x^{2}=4 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 4\) and \(x = -4\)

Step 1: Rewrite the Equation in Standard Form

The given equation is \(20 - x^2 = 4\). Let's rewrite this in standard form by subtracting 4 from both sides. This gives us the equation as \(-x^2 + 20 - 4 = 0\). Simplify to \(-x^2 + 16 = 0\).

Step 2: Solve the Equation

We can solve the equation by using the square root property. Start by isolating \(x^2\), so rewrite the equation as \(x^2 = 16\). Then take the square root of both sides. Remember to consider both positive and negative roots. This gives \(x = \pm \sqrt{16}\).

Step 3: Final Solution

Calculating the square root of 16 gives \(x = \pm 4\). Therefore the equation has two solutions: \(x=4\) and \(x=-4\).

Key Concepts

Square Root Property

The square root property is a useful tool in solving quadratic equations, especially when the equation is already set to a perfect square. It allows you to take the square root of both sides of the equation, helping to isolate the variable. When applying this property, it is essential to remember that taking the square root introduces two possible solutions: one positive and one negative.

• Consider the equation: \( x^2 = k \).
• To solve it, you take the square root of both sides, yielding \( x = \pm \sqrt{k} \).

This shows that both the positive and negative roots must be considered as potential solutions. This concept is particularly helpful when dealing with equations like the quadratic one from our exercise, where the variable is squared.

Standard Form of Quadratic Equation

Quadratic equations are commonly expressed in their standard form, which is \( ax^2 + bx + c = 0 \). This format makes it easier to apply various mathematical methods for solving, such as factoring, completing the square, or using the quadratic formula.

In the standard form:

• \(a\), \(b\), and \(c\) are constants.
• \(a\) should not be zero, as this would make the equation linear instead of quadratic.

For example, the given problem was initially \(20 - x^2 = 4\). Rewriting it in the standard form, we get \(-x^2 + 16 = 0\). This makes it more recognizable as a quadratic equation and prepares it for applying the appropriate solution strategies.

Radical Expressions

Radical expressions involve roots, most commonly the square root. In equations, radical expressions can appear when isolating terms that involve variables squared or higher powers. Working with radicals involves certain rules that help simplify expressions and solve equations.

• Remember the principal square root symbol (\( \sqrt{} \)) represents the positive root.
• Radical expressions can sometimes remain unsimplified, especially if the number is not a perfect square. For example, \( \sqrt{17} \) remains as is because 17 is not a perfect square.

When solving quadratic equations, you often use radical expressions to state the solutions. As seen in the exercise, the calculation \( x = \pm \sqrt{16} \) results in two integer solutions, \(x = 4\) and \(x = -4\), because 16 is a perfect square.