Problem 40 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$x 2-18=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 3\sqrt{2}\) and \(x = -3\sqrt{2}\).

Step 1: Identify the Coefficients

The given equation is already in the quadratic form: \(x^2 - 18 = 0\). We need to identify the coefficients. Here, \(a = 1\), \(b = 0\), and \(c = -18\).

Step 2: Write the Quadratic Formula

The quadratic formula is given by: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]

Step 3: Substitute the Coefficients into the Formula

Substitute \(a = 1\), \(b = 0\), and \(c = -18\) into the quadratic formula: \[ x = \frac{{-0 \pm \sqrt{{0^2 - 4 \cdot 1 \cdot (-18)}}}}{2 \cdot 1} \] This simplifies to: \[ x = \frac{{\pm \sqrt{{72}}}}{2} \]

Step 4: Simplify the Expression

Next, we simplify \(\sqrt{72}\):\[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \] Now substitute back:\[ x = \frac{{\pm 6\sqrt{2}}}{2} \]

Step 5: Obtain Final Result

Divide the terms by 2 to find:\[ x = \pm 3\sqrt{2} \] Thus, the solutions are \(x = 3\sqrt{2}\) and \(x = -3\sqrt{2}\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They are polynomial equations of the second degree, typically written in the form of \(ax^2 + bx + c = 0\). Recognizing this form is crucial when solving these equations. The goal is to find the value of \(x\) that satisfies the equation. Such equations are called "quadratic" because "quad" refers to a square, and they involve \(x^2\).
Quadratic equations can have:

• Two real and distinct roots
• One real root (repeated)
• No real roots when the solutions are complex numbers

The nature of the roots depends on the discriminant, \(b^2 - 4ac\). This part of the quadratic formula determines the type of solutions the quadratic equation will have. Understanding this will help when you apply the quadratic formula to solve it.

Coefficients

Coefficients are numerical or constant terms that multiply the variables in an equation. For instance, in the quadratic equation \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are the coefficients. Each coefficient plays a specific role:

• \(a\) is the coefficient of \(x^2\), often impacting the width and direction of the parabola graphed from the equation.
• \(b\) is the coefficient of \(x\), influencing the position of the vertex along the x-axis.
• \(c\) is the constant term, affecting where the parabola intersects the y-axis.

In the given example, we identified \(a = 1\), \(b = 0\), and \(c = -18\). Knowing how to identify and assign these values is essential for plugging into the quadratic formula accurately.

Simplifying Radicals

Simplifying radicals is a step often necessary when solving quadratic equations using the quadratic formula. The solution often includes a square root, \(\sqrt{b^2 - 4ac}\), known as the discriminant. Simplifying this radical can clarify the root values and make the solution more exact.
Here's how to simplify a radical:

• Identify if you can factor the number inside the square root into perfect squares.
• Break it down into \(\sqrt{ ext{perfect square} \times ext{other term}}\).
• Use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
• Simplify further if possible.

For example, simplifying \(\sqrt{72}\) involves recognizing that \(72 = 36 \times 2\), allowing us to write \(\sqrt{72} = 6\sqrt{2}\). Making these simplifications is crucial for finding precise solutions to quadratic equations.