Problem 43 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$-2 d^{2}-5 d+19=0$$

Step-by-Step Solution

Answer

The roots of the equation \(-2 d^{2}-5 d+19 = 0\) are \(d=\frac{5+\sqrt{177}}{-4}\) and \(d=\frac{5-\sqrt{177}}{-4}\)

Step 1: Identify the coefficients

From the quadratic equation, identify the coefficients of \(d^{2}\), \(d\) and the constant as \(a = -2\), \(b = -5\) and \(c = 19\), respectively.

Step 2: Substitute the coefficients into the quadratic formula

Substitute the coefficients \(a\), \(b\), and \(c\) into the quadratic formula \(d=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Thus the formula becomes \(d=\frac{-(-5)\pm\sqrt{(-5)^{2}-4*(-2)*19}}{2*(-2)}\).

Step 3: Simplify the equation

Simplify the equation further to get the roots. Hence, we get the roots as \(d=\frac{5\pm\sqrt{25+152}}{-4}\). Now the equation simplifies to \(d=\frac{5\pm\sqrt{177}}{-4}\). Therefore, the roots of the quadratic equation are \(d=\frac{5+\sqrt{177}}{-4}\) and \(d=\frac{5-\sqrt{177}}{-4}\).

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool in algebra used to find the solutions or "roots" of a quadratic equation. A quadratic equation is generally written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\). The quadratic formula itself is expressed as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The beauty of this formula is that it provides a direct way to solve any quadratic equation, no matter how complex it might seem. To use the quadratic formula, you need to correctly identify the coefficients \(a\), \(b\), and \(c\) from your specific quadratic equation. Once these are identified, substitute them into the quadratic formula to find the values of \(x\) that satisfy the equation.

• \(b^2 - 4ac\) is called the discriminant and it determines the nature of the roots.
• \(\pm\) symbol indicates that there are generally two solutions.

This formula is particularly useful when factorizing doesn't work or when the equation is not easily simplified by other means.

Roots of Equations

The roots of a quadratic equation are the values of \(x\) that make the equation equal to zero. In other words, they are the solutions to the equation \(ax^2 + bx + c = 0\). The quadratic formula helps us find these roots by solving for \(x\).
The roots can vary in nature depending on the discriminant, \(b^2 - 4ac\):

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root (also called a repeated or double root).
• If the discriminant is negative, the roots are complex or imaginary numbers.

When using the quadratic formula, after calculating \(\sqrt{b^2 - 4ac}\), you apply the "plus" and "minus" parts of the \(\pm\) to find each root. This gives the equation two solutions in most cases. Understanding the nature of these roots is important for graphing quadratic functions and analyzing real-world problems.

Coefficients in Algebra

In algebra, coefficients are the numbers attached to variables in mathematical expressions. In the context of quadratic equations, the coefficients are the constants \(a\), \(b\), and \(c\) in the equation \(ax^2 + bx + c = 0\). Each coefficient plays a significant role in determining the shape and position of the parabola that the quadratic equation represents.

• \(a\): The coefficient of \(x^2\), determines the "width" and "direction" of the parabola. If \(a\) is positive, the parabola opens upwards. If it's negative, the parabola opens downwards.
• \(b\): The coefficient of \(x\), influences the "tilt" or "balance" of the parabola, impacting where the vertex lies along the x-axis.
• \(c\): The constant term affects the "vertical shift" of the parabola, being the point where the curve crosses the y-axis when \(x = 0\).

Grasping how these coefficients affect the equation's graph can deepen your understanding of quadratic equations beyond mere calculation. They provide insights into the symmetric properties and the direction of the curve's opening.