Problem 20 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$2 d(d-2)=-7$$

Step-by-Step Solution

Answer

The quadratic equation has no real roots.

Step 1: Rewrite the Equation

Start by expanding the left side of the given equation. Distribute the term to get: \[2d(d - 2) = 2d^2 - 4d\]Then substitute this into the equation to get:\[2d^2 - 4d = -7\]

Step 2: Rearrange to Standard Form

Move all the terms to one side of the equation to set it to 0:\[2d^2 - 4d + 7 = 0\]

Step 3: Identify Coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c = 0\):\[a = 2, \, b = -4, \, c = 7\]

Step 4: Apply the Quadratic Formula

Use the quadratic formula:\[d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute the identified values:\[d = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 2 \times 7}}{2 \times 2}\]

Step 5: Calculate the Discriminant

First, calculate the discriminant \(b^2 - 4ac\):\[(-4)^2 - 4 \times 2 \times 7 = 16 - 56 = -40\]Since the discriminant is negative, the quadratic equation has no real roots.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation, which is an equation that can be written in the form \[ ax^2 + bx + c = 0 \]. The formula is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
It allows us to solve any quadratic equation by substituting the coefficients \(a\), \(b\), and \(c\) into the formula. A key advantage of the quadratic formula is that it provides the roots directly, regardless of whether they are real or complex.

• The term \(\pm\) indicates that there can be two solutions: one with addition and the other with subtraction.
• The expression under the square root, \(b^2 - 4ac\), is known as the discriminant, which helps determine the nature of the roots.

It is essential to understand how to use this formula effectively, as it applies to all types of quadratic equations.

Standard Form

Before we apply the quadratic formula, we need to ensure that the quadratic equation is in standard form. The standard form of a quadratic equation is expressed as:\[ ax^2 + bx + c = 0 \].
In this form, the term \(ax^2\) represents the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term. To convert an equation to this form, you must arrange all terms such that one side of the equation equals zero.

• For example, if you have an equation like \(2d(d - 2) = -7\), you need to expand and rearrange the terms to get \( 2d^2 - 4d + 7 = 0 \).
• Once in standard form, you can clearly identify the coefficients needed for the quadratic formula.

Standard form makes it easier to apply the quadratic formula effectively and find the roots of the equation.

Coefficients

Coefficients play a crucial role in quadratic equations and directly influence the solutions derived using the quadratic formula. In the context of the standard form equation \( ax^2 + bx + c = 0 \), the coefficients are the numbers associated with each term.
They provide the necessary values to substitute into the quadratic formula:

• \(a\) is the coefficient of \(x^2\), representing the quadratic term.
• \(b\) is the coefficient of \(x\), representing the linear term.
• \(c\) is the constant term, which is the standalone number without an \(x\).

Identifying these coefficients accurately is vital, as incorrect values can lead to incorrect solutions. For instance, in the equation \(2d^2 - 4d + 7 = 0\), the coefficients are \(a = 2\), \(b = -4\), and \(c = 7\). Substitution of these correct values into the quadratic formula provides meaningful results.

Discriminant

The discriminant is a key component of the quadratic formula that determines the nature of the roots for a quadratic equation. It is the part of the formula given by \( b^2 - 4ac \).
The discriminant provides insight into how many and what type of roots you can expect from the quadratic equation.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root, sometimes called a repeated or double root.
• If the discriminant is negative, as in the problem where \((-4)^2 - 4 \times 2 \times 7 = -40\), there are no real roots; rather, the roots are complex.

Understanding the discriminant helps predict the nature of the quadratic's solutions before solving, providing an initial check for the solution process.