Problem 129 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(2 x^{2}+12 x-3=0\)

Step-by-Step Solution

Answer

The solutions are \( x = -3 + \frac{\sqrt{42}}{2} \) and \( x = -3 - \frac{\sqrt{42}}{2} \).

Step 1: Identify coefficients

First, identify the coefficients for the quadratic equation in the form of \( ax^{2} + bx + c = 0 \). Here, \( a = 2 \), \( b = 12 \), and \( c = -3 \).

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 values into the formula

Substitute \( a = 2 \), \( b = 12 \), and \( c = -3 \) into the formula: \( x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2} \).

Step 4: Simplify inside the square root

Calculate the value inside the square root: \( 12^2 - 4 \cdot 2 \cdot (-3) = 144 + 24 = 168 \). So, the formula becomes: \( x = \frac{-12 \pm \sqrt{168}}{4} \).

Step 5: Simplify the square root

Simplify \( \sqrt{168} \). \( \sqrt{168} = \sqrt{4 \cdot 42} = \sqrt{4} \cdot \sqrt{42} = 2 \sqrt{42} \).

Step 6: Final simplification

Substitute \( \sqrt{168} = 2 \sqrt{42} \) back into the formula: \( x = \frac{-12 \pm 2 \sqrt{42}}{4} \). Simplify further: \( x = \frac{-12 + 2 \sqrt{42}}{4} = \frac{-12 - 2 \sqrt{42}}{4} \).

Step 7: Express in simplest form

Divide both terms in the numerator by 4: \( x = \frac{-12}{4} + \frac{2 \sqrt{42}}{4} = -3 + \frac{\sqrt{42}}{2} \) and \( x = \frac{-12}{4} - \frac{2 \sqrt{42}}{4} = -3 - \frac{\sqrt{42}}{2} \).

Key Concepts

solving quadratic equations

A quadratic equation is an equation of the form \[ax^{2} + bx + c = 0\]To find the solutions of a quadratic equation, you can use the Quadratic Formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this formula,

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

The '±' symbol in the formula indicates that there will be two solutions for the quadratic equation: one with addition ( + ) and one with subtraction ( - ). Follow these steps to apply the Quadratic Formula: Identify the coefficients in the given equation. Substitute the coefficients into the Quadratic Formula. Simplify the expression under the square root. Calculate the solutions.

simplifying square roots

When solving quadratic equations, you might need to simplify square roots. For example, \(\sqrt{168}\). To simplify a square root:

• Factor the number inside the square root into its prime factors.
• Look for perfect square factors.
• Rewrite the square root as a product of square roots.
• Simplify the square root of perfect squares.

In our example, \(\sqrt{168} = \sqrt{4 \cdot 42} = \sqrt{4} \cdot \sqrt{42} = 2 \sqrt{42}\) This makes it easier to proceed with the solving process. Simplifying square roots helps in simplifying your final answers.

coefficients in algebra

Coefficients are the numerical factors in the terms of an equation. In the quadratic equation \(2x^{2}+12x-3=0\),

• The coefficient of \(x^{2}\) is 2.
• The coefficient of \(x\) is 12.
• The constant term is -3.

Identifying the coefficients correctly is crucial to applying the Quadratic Formula. Misidentifying them can result in completely incorrect solutions. Whenever you're solving equations, always double-check the coefficients before plugging them into formulas. This makes solving the equation accurately and simplification easier.