Problem 54 - Elementary Algebra

Question

\(d^{2}+10 d+16=0\)

Step-by-Step Solution

Answer

The roots are \( d = -2 \) and \( d = -8 \).

Step 1 - Identify the form of the equation

Recognize that the equation is a quadratic equation in the form of: \[ ax^2 + bx + c = 0 \] Here, \( a = 1 \), \( b = 10 \), and \( c = 16 \).

Step 2 - Calculate the discriminant

Calculate the discriminant by using the formula: \[ \text{Discriminant} = b^2 - 4ac \] Substitute the values of \( a \), \( b \), and \( c \) into the equation: \[ \text{Discriminant} = 10^2 - 4(1)(16) \] \[ \text{Discriminant} = 100 - 64 \] \[ \text{Discriminant} = 36 \]

Step 3 - Solve for the roots using the quadratic formula

Use the quadratic formula to find the roots: \[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values of \( a \), \( b \), and \( \text{Discriminant} \): \[ d = \frac{-10 \pm \sqrt{36}}{2(1)} \] \[ d = \frac{-10 \pm 6}{2} \]

Step 4 - Calculate the individual roots

Calculate the two possible values for \( d \): For the positive square root:\[ d = \frac{-10 + 6}{2} = \frac{-4}{2} = -2 \] For the negative square root:\[ d = \frac{-10 - 6}{2} = \frac{-16}{2} = -8 \]

Step 5 - State the final roots

The solutions to the quadratic equation are:\[ d = -2 \quad \text{and} \quad d = -8 \]

Key Concepts

quadratic formula

One common method to solve quadratic equations is by using the quadratic formula. This formula provides the solutions (or roots) of any quadratic equation, which has the general form \[ ax^2 + bx + c = 0 \] The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula is derived from completing the square on the general quadratic equation and it is a reliable tool for finding the values of \( x \) that satisfy the equation.

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

All you need is to plug in the values of \( a \), \( b \), and \( c \) into the formula and solve for \( x \). This is exactly what we did in the exercise above to find the values of \( d \).

discriminant

An important part of the quadratic formula is the discriminant. The discriminant is the expression under the square root in the quadratic formula: \[ \Delta = b^2 - 4ac \] The value of the discriminant determines the nature of the roots of the quadratic equation.

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), the quadratic equation has exactly one real root (repeated root).
• If \( \Delta < 0 \), the quadratic equation has two complex (non-real) roots.

In the original exercise, we calculated the discriminant as \[ \Delta = 10^2 - 4(1)(16) = 36 \] Since 36 is greater than zero, the quadratic equation has two distinct real roots, which we found to be \( d = -2 \) and \( d = -8 \). The discriminant is a quick way to understand the nature of the solutions without fully solving the equation.

roots of a quadratic equation

The solutions to a quadratic equation, also known as the roots, are the values of \( x \) (or any other variable) that satisfy the equation. These roots can be found using various methods like factoring, completing the square, or using the quadratic formula.
In the exercise given, we used the quadratic formula to find the roots. We started by calculating the discriminant and then substituted it back into the formula. The solutions we found were \[ d = \frac{-10 + 6}{2} = -2 \] and \[ d = \frac{-10 - 6}{2} = -8 \] These values are the points where the quadratic equation \[ d^2 + 10d + 16 = 0 \] intersects the x-axis.
Roots are fundamental in understanding the behavior of quadratic functions. They provide valuable information about the function's graph, such as where it crosses the x-axis and the symmetry of the function around its vertex.