Problem 130 - College Algebra

Question

Solve the equation: x^{2}+2 \sqrt{3} x-9=0

Step-by-Step Solution

Answer

The solutions to this quadratic equation are \( x = \sqrt{3} \) and \( x = -3\sqrt{3} \).

Step 1: Identifying Coefficients

In the equation \( x^{2}+2 \sqrt{3} x-9=0 \), the coefficient 'a' is 1, 'b' is \(2 \sqrt{3}\), and 'c' is -9.

Step 2: Calculating Under the Square Root in Quadratic Formula

Utilize the formula \(b^{2}-4ac\). Substituting values we get \((2 \sqrt{3})^{2}-4*1*(-9)\), which simplifies to \(12+36 = 48\).

Step 3: Using Quadratic Formula

Now substitute these values in the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \) and calculate the value of 'x'. This would yield \( x = \frac{-2 \sqrt{3} \pm \sqrt{48}}{2} \). To simplify this, we calculate \( x1 = \frac{-2 \sqrt{3} + \sqrt{48}}{2} \), and \( x2 = \frac{-2 \sqrt{3} - \sqrt{48}}{2} \). Simplifying further gives the solutions \( x = \sqrt{3} \) and \( x = -3\sqrt{3} \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The formula provides the solution for \( x \) in terms of the coefficients of the quadratic equation:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

The quadratic formula itself is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula gives two solutions for \( x \), corresponding to the \( + \) and \( - \) operations, known as the roots of the equation. Using this formula, you can find the exact solutions of any quadratic equation, assuming the calculation under the square root (the discriminant) is non-negative.

Solving Quadratic Equations

To solve a quadratic equation using the quadratic formula, follow these steps:

• Identify the coefficients \( a \), \( b \), and \( c \) from the equation \( ax^2 + bx + c = 0 \).
• Calculate the discriminant \( b^2 - 4ac \).
• Substitute \( a \), \( b \), and the discriminant into the quadratic formula.
• Compute the solutions for \( x \) using both the positive and negative values in the formula.

In the given exercise, the equation is \( x^2 + 2\sqrt{3}x - 9 = 0 \). Using the quadratic formula, we substitute \( a = 1 \), \( b = 2\sqrt{3} \), and \( c = -9 \). Solving this, you find the roots \( x = \sqrt{3} \) and \( x = -3\sqrt{3} \). These represent the points where the quadratic function crosses the x-axis.

Radicals in Algebra

Radicals are expressions that involve roots, such as square roots, cube roots, etc. In algebra, radicals are used to simplify expressions and solve equations. When you encounter a radical like \( \sqrt{48} \), it can often be simplified. To simplify \( \sqrt{48} \):

• Recognize that \( 48 = 16 \times 3 \)
• This allows you to simplify \( \sqrt{48} \) as \( \sqrt{16} \times \sqrt{3} \) which becomes \( 4\sqrt{3} \)

Simplifying radicals can make it easier to solve equations and understand the nature of the solutions. In the quadratic formula, simplifying the square root term is a crucial step in expressing the roots of the equation in their simplest form. This process ensures clarity when interpreting the solutions of quadratic equations, whether they have real or complex roots.