Problem 21 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations. If there are no real roots, state this as the answer. \(x(2 x-1)=-3\)

Step-by-Step Solution

Answer

The equation has no real roots due to a negative discriminant.

Step 1: Expand the Equation

Start by expanding the left side of the equation. The equation is given as \(x(2x - 1) = -3\). Distribute the \(x\) to both terms inside the parenthesis: \(2x^2 - x = -3\).

Step 2: Rearrange into Standard Quadratic Form

Rearrange the equation into the standard quadratic form \(ax^2 + bx + c = 0\). Adding \(3\) to both sides gives \(2x^2 - x + 3 = 0\).

Step 3: Use the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Identify \(a = 2\), \(b = -1\), and \(c = 3\).

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). Substitute the values: \((-1)^2 - 4(2)(3) = 1 - 24 = -23\).

Step 5: Interpret the Discriminant

Since the discriminant \(-23\) is negative, the equation has no real roots. The roots are complex numbers, but since the exercise specifies real roots only, we conclude there are none.

Key Concepts

Discriminant

When dealing with quadratic equations, a critical component in solving them is the discriminant. The discriminant is part of the quadratic formula and is represented as \(b^2 - 4ac\). This value helps to determine the nature and number of roots for the quadratic equation.

Here's why the discriminant is important:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the quadratic equation does not have real roots; instead, it has two complex conjugate roots.

Using the discriminant is a quick way to check the type of roots without necessarily solving the entire quadratic equation. In our original problem, we calculated the discriminant as \(-23\), indicating no real roots because it is negative.

Quadratic Formula

The quadratic formula is a universal tool for finding the roots of quadratic equations. It's especially useful when factoring is too complex or impossible, or when other methods like completing the square aren't suitable. The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows us to solve any quadratic equation written in the form \(ax^2 + bx + c = 0\). By identifying the coefficients (\(a\), \(b\), and \(c\)), we can substitute them into the quadratic formula to find the roots.

The expression under the square root is the discriminant \(b^2-4ac\), which influences the nature of the roots as discussed. Understanding how to properly apply the quadratic formula can turn what appears to be a complex problem into a straightforward computation.

Real Roots

Real roots occur when a quadratic equation's solutions are actual numbers on the real number line. They can be visually represented as where the graph of the quadratic equation intersects the \(x\)-axis.

Here’s how the discriminant we discussed earlier ties in:

• A positive discriminant leads to two real roots, meaning the parabola crosses the \(x\)-axis at two distinct points.
• A zero discriminant leads to one real root, meaning the vertex of the parabola touches the \(x\)-axis.
• A negative discriminant means there are no real roots, and the parabola does not intersect the \(x\)-axis at all.

In our example equation, since the discriminant was \(-23\), there were no real roots, further confirming that the parabola does not touch or cross the \(x\)-axis.

Complex Numbers

When quadratic equations have no real roots due to a negative discriminant, the roots are considered complex numbers. These are numbers that include a real component and an imaginary component, typically represented in the form \(a + bi\), where \(i\) is the imaginary unit (\(\sqrt{-1}\)).

Complex numbers are the solution when you square root a negative number. In our exercise problem, since \(b^2 - 4ac = -23\), the roots of the equation are complex. Even though complex numbers might seem abstract, they extend the concept of numbers to solve equations without real solutions, providing a complete view of the mathematical landscape. Understanding them opens up more advanced fields in mathematics and engineering.