Problem 26 - Beginning and Intermediate Algebra

Question

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ -3 x(x+2)=-4 $$

Step-by-Step Solution

Answer

x = -1 \pm \frac{\sqrt{21}}{3}

Step 1: Expand and Simplify the Equation

First, expand the left-hand side by distributing -3 inside the parentheses: -3x(x + 2) = -4 This results in: -3x^2 - 6x = -4

Step 2: Move All Terms to One Side

Add 4 to both sides to set the equation to zero: -3x^2 - 6x + 4 = 0

Step 3: Identify Coefficients

Identify coefficients a, b, and c from the standard quadratic form ax^2 + bx + c = 0In this case, a = -3, b = -6, and c = 4

Step 4: Apply the Quadratic Formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plug in the values a = -3, b = -6, and c = 4 into the formula: \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(-3)(4)}}{2(-3)} \] Which simplifies to: \[ x = \frac{6 \pm \sqrt{36 + 48}}{-6} \]

Step 5: Solve Inside the Square Root

Simplify inside the square root: \[ x = \frac{6 \pm \sqrt{84}}{-6} \] Since \( \sqrt{84} \) can be simplified to \( 2 \sqrt{21} \), the equation becomes: \[ x = \frac{6 \pm 2 \sqrt{21}}{-6} \]

Step 6: Simplify the Expression

Simplify the entire fraction: \[ x = -1 \pm \frac{\sqrt{21}}{3} \]

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). To solve them, one efficient method is using the quadratic formula. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation. Using the quadratic formula simplifies the process and ensures accurate results.
When applying the quadratic formula:

• Identify the coefficients (\(a\), \(b\), and \(c\)) from the equation.
• Plug these coefficients into the formula.
• Simplify the expression under the square root (discriminant).
• Compute the final solutions.

This step-by-step method helps in systematically breaking down the problem, making it easier to solve.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and performing arithmetic operations to make the expression as simple as possible. For quadratic equations, this usually involves:

• Expanding parentheses.
• Combining like terms.
• Rewriting the equation in standard form \(ax^2 + bx + c = 0\).

In the given exercise, we started with: \[-3x(x + 2) = -4\]
After expanding, it becomes: \[-3x^2 - 6x = -4\]
Next, we need to move all terms to one side to set the equation to zero: \[-3x^2 - 6x + 4 = 0\]
Simplifying the equation in this manner is crucial for accurately applying the quadratic formula.

Quadratic Equation Coefficients

Coefficients in a quadratic equation are the numerical constants in front of the variables. For the standard form \(ax^2 + bx + c = 0\), the coefficients are critical as they play a key role in solving the equation.

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

In our example, after simplifying, the equation: \[-3x^2 - 6x + 4 = 0\] has:

• Coefficient \(a = -3\).
• Coefficient \(b = -6\).
• Constant \(c = 4\).

Understanding these coefficients is crucial for correctly applying the quadratic formula and solving the equation.