Problem 52 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 3 x^{2}+14 x=-8 $$

Step-by-Step Solution

Answer

The solutions are \( x = -\frac{2}{3} \) and \( x = -4 \).

Step 1: Move All Terms to One Side

Begin by moving all terms to one side of the equation to set it to zero. This gives us:\[ 3x^2 + 14x + 8 = 0 \]

Step 2: Identify Quadratic Coefficients

Identify the coefficients for the quadratic equation \( ax^2 + bx + c = 0 \). In this case, \( a = 3 \), \( b = 14 \), and \( c = 8 \).

Step 3: Use the Quadratic Formula

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) will help us find the roots. Substitute the coefficients \( a \), \( b \), and \( c \) into the formula: \[ x = \frac{-14 \pm \sqrt{(14)^2 - 4 \times 3 \times 8}}{2 \times 3} \]

Step 4: Simplify the Discriminant

Compute the discriminant \( b^2 - 4ac \):\[ 14^2 - 4 \times 3 \times 8 = 196 - 96 = 100 \]

Step 5: Calculate the Square Root

Find the square root of the discriminant:\[ \sqrt{100} = 10 \]

Step 6: Solve for Roots

Substitute \( \sqrt{100} \) back into the quadratic formula:\[ x = \frac{-14 \pm 10}{6} \]This results in two potential solutions:1. \( x = \frac{-14 + 10}{6} = \frac{-4}{6} = -\frac{2}{3} \)2. \( x = \frac{-14 - 10}{6} = \frac{-24}{6} = -4 \)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of quadratic equations. Quadratic equations are in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This equation allows you to calculate the values for \( x \) that make the quadratic equation true. The \(-b \pm\) part indicates two possible solutions, representing the two potential roots a quadratic equation can have. It splits the solution into two parts, each being calculated based on the value of the discriminant. The quadratic formula is an essential tool because it systematically provides an exact answer to any quadratic equation, regardless of whether the roots are real or complex. Understanding how to use this formula is key when dealing with quadratic equations, especially in contexts where factoring is challenging or impossible.

• Substitute the coefficients in the formula.
• Calculate the discriminant within the square root.
• Proceed with solving the equation given the two options from \( \pm \).

Discriminant

The discriminant is a crucial part of the quadratic formula and can be found inside the square root: \( b^2 - 4ac \). It helps determine the nature of the roots of the quadratic equation. This step is important because just computing it can tell you about the roots without having to solve the entire formula. Here's what the discriminant can reveal:

• If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, which means that both roots are equal.
• If \( b^2 - 4ac < 0 \), the equation has no real roots but two complex roots.

In the example provided, the discriminant is calculated as \( 14^2 - 4 \times 3 \times 8 = 100 \). Since 100 is greater than 0, this indicates there are two distinct real roots for the equation. This aspect of quadratic equations allows you to assess whether traditional solving methods or further analysis for complex roots are required.

Roots of Quadratic Equation

Every quadratic equation has roots, which are the solutions to the equation when set to zero. In the context of plotting a quadratic function, these roots represent the points where the parabola intersects the x-axis. The quadratic formula yields these points, which can provide valuable insights into the behavior of the graph. Finding the roots \( x = \frac{-14 \pm 10}{6} \) gives us two solutions:

• \( x = \frac{-14 + 10}{6} = -\frac{2}{3} \), which means the parabola crosses the x-axis at \( x = -\frac{2}{3} \).
• \( x = \frac{-14 - 10}{6} = -4 \), indicating another intersection at \( x = -4 \).

Understanding these roots helps to visualize the quadratic function. From the equation \( 3x^2 + 14x + 8 = 0 \), the roots tell us where the graph of this quadratic function touches or crosses the horizontal axis, effectively describing the zeros of the function. This understanding is critical in algebra and is especially useful in advanced topics like calculus and physics, where finding these points is necessary to interpret or solve real-world problems.