Problem 65 - Algebra and Trigonometry

Question

Solve equation using the quadratic formula. $$ x^{2}+8 x+15=0 $$

Step-by-Step Solution

Answer

The solutions of the equation are \(x=-5\) and \(x=-3.5\).

Step 1: Identify coefficients

From the quadratic equation \(x^2 + 8x + 15 = 0\), the coefficients can be identified as \(a = 1\), \(b = 8\), and \(c = 15\).

Step 2: Calculate the discriminant

The discriminant for a quadratic equation \(\(b^2 - 4ac)\). For our equation, it will be \(64 - 4(1)(15) = 64 - 60 = 4\).

Step 3: Apply the quadratic formula

Substitute the coefficients and discriminant into the quadratic formula \(x = \[-b ± √(b^2 - 4ac) / 2a\]\). So we can get two solutions: \(x1 = \[-8+√4]/2 = -7/2 = -3.5\]\) and \(x2 = \[-8 - √4]/2 = -10/2 = -5\]\). So the roots of the equation are \(-3.5\) and \(-5\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written in the form

• \(ax^2 + bx + c = 0\)

Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). This is crucial because if \(a\) were zero, the equation would be linear, not quadratic. Quadratic equations describe a wide range of phenomena, from projectiles to areas in mathematics.

Solving these equations involves finding the value of \(x\) that makes the equation true. This is done using the quadratic formula, factoring, or completing the square. Each method has its benefits, but the quadratic formula is universal and works with any quadratic equation.

Whenever you're confronted with a quadratic equation, rewriting it in the standard form if it's not already is a vital first step.

The Role of the Discriminant in Quadratic Equations

The discriminant plays a key role in understanding the nature of the roots for any given quadratic equation. It is given by the expression

• \(b^2 - 4ac\)

Here, \(a\), \(b\), and \(c\) are the coefficients in the quadratic equation \(ax^2 + bx + c = 0\).

The value of the discriminant tells us several things:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, the quadratic equation has exactly one real root, indicating that the parabola touches the x-axis at one point.
• If the discriminant is negative, the quadratic equation has two complex roots; it does not intersect the x-axis.

In our example, the discriminant is 4, which is positive. This means we will have two distinct real roots. Understanding the discriminant is essential for predicting the behavior of quadratic equations.

Finding the Roots of the Equation

When solving a quadratic equation, you are trying to find the roots, which are the solutions for \(x\) that satisfy \(ax^2 + bx + c = 0\). The quadratic formula

• \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

makes this process manageable regardless of the equation's complexity.

Using the quadratic formula involves:

• Calculating the discriminant \(b^2 - 4ac\)
• Substituting \(-b\), the discriminant’s square root, and \(a\) into the formula

This results in two possible values for \(x\), labeled as \(x_1\) and \(x_2\). In the given example, substitution leads to roots

• \(x_1 = -3.5\)
• \(x_2 = -5\)

These roots indicate where the quadratic equation intersects the x-axis. Finding the roots accurately is crucial for plotting the equation or understanding its real-world implications.

Coefficients in Quadratic Equations

In any quadratic equation of the form

• \(ax^2 + bx + c = 0\)

\(a\), \(b\), and \(c\) are known as coefficients.

Understanding these coefficients is essential as they determine the shape and position of the parabola represented by the equation. Here's how each impacts the equation:

• The coefficient \(a\) affects the concavity (direction it opens) and the width of the parabola. A positive \(a\) makes it open upwards, while a negative \(a\) opens it downwards.
• The coefficient \(b\) influences the shift and tilt of the parabola along the x-axis.
• The coefficient \(c\) is the constant term and represents the y-intercept, where the parabola crosses the y-axis.

In the example

• \(a = 1\)
• \(b = 8\)
• \(c = 15\)

These coefficients are the foundational elements needed to solve the quadratic equation using any method, especially the quadratic formula.