Problem 22 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations. Round solutions to the nearest hundredth. If there are no real roots, state this. $$6 w-15=3 w^{2}$$

Step-by-Step Solution

Answer

The equation has no real roots.

Step 1: Write the Equation in Standard Form

The given equation is \(6w - 15 = 3w^2\). Rewrite it in standard quadratic form by setting everything to one side: \(3w^2 - 6w + 15 = 0\).

Step 2: Identify Coefficients

Identify the coefficients of the quadratic equation \(ax^2 + bx + c = 0\). Here, \(a = 3\), \(b = -6\), and \(c = 15\).

Step 3: Calculate the Discriminant

The discriminant \(D\) is given by the formula \(b^2 - 4ac\). Substitute the values: \((-6)^2 - 4\cdot3\cdot15 = 36 - 180 = -144\).

Step 4: Analyze the Discriminant

Since the discriminant \(D = -144\) is less than zero, the quadratic equation has no real roots. This means there are no real solutions to the equation.

Key Concepts

Discriminant

The discriminant of a quadratic equation is a key element in determining the nature of the roots of the equation. It is denoted by the letter "\(D\)" and calculated using the formula:\[ D = b^2 - 4ac \]Here, \(a\), \(b\), and \(c\) are the coefficients from the standard form of a quadratic equation \(ax^2 + bx + c = 0\). The discriminant provides valuable information:

• If \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, also known as a repeated or double root.
• If \(D < 0\), no real roots exist; the roots are complex or imaginary.

In this particular exercise, after calculating the discriminant as \(-144\), it is clear that since \(D < 0\), there are no real solutions. This makes the concept of the discriminant crucial for solving and understanding quadratic equations.

Real Roots

Real roots are the solutions to a quadratic equation where the values satisfy the equation such that the outputs are real numbers, rather than complex numbers. Finding real roots involves solving the quadratic equation, typically through factoring, completing the square, or using the quadratic formula, which is:\[ x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a} \]The term under the square root sign, \(b^2 - 4ac\), is the discriminant, which is vital in determining whether real roots exist. Real roots only occur if the discriminant is non-negative:
- Two distinct real roots if \(D > 0\).- One real root if \(D = 0\), meaning the parabola touches the x-axis at a single point.In the exercise at hand, since the discriminant was found to be \(-144\), it indicates there are no real roots, hence no real numbers that satisfy the equation \(3w^2 - 6w + 15 = 0\). Instead, the solutions would be complex numbers.

Standard Form Quadratic Equation

A standard form quadratic equation is structured as:\[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). This form is essential for analyzing and solving quadratic equations, as it allows for the identification of coefficients used in various methods, such as using the quadratic formula or determining the discriminant.
The problem given, \(6w - 15 = 3w^2\), is transformed into its standard form by rearranging to:\[ 3w^2 - 6w + 15 = 0 \]This arrangement not only simplifies calculations of solutions but also aids in understanding the relationship between the coefficients and the equation's behavior. Once in standard form, identifying \(a = 3\), \(b = -6\), and \(c = 15\) becomes straightforward, which is crucial for any subsequent steps, such as calculating the discriminant or using the quadratic formula to determine roots. This format standardizes the approach to solving and analyzing quadratic equations effectively.