Problem 19 - 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. \(-3 x^{2}+11 x-5=0\)

Step-by-Step Solution

Answer

The equation has two real roots: approximately 0.53 and 3.14.

Step 1: Identify the Equation Type

The given equation \[-3x^2 + 11x - 5 = 0\]is a quadratic equation of the form \(ax^2 + bx + c = 0\). Here, \(a = -3\), \(b = 11\), and \(c = -5\).

Step 2: Determine Discriminant

The discriminant of a quadratic equation is given by \[D = b^2 - 4ac\]Substitute the given values:\[D = 11^2 - 4(-3)(-5) = 121 - 60 = 61\]

Step 3: Check Discriminant Sign

Since the discriminant \(D = 61\) is greater than zero, the quadratic equation has two distinct real roots.

Step 4: Use the Quadratic Formula

The quadratic formula to find the roots is \[x = \frac{-b \pm \sqrt{D}}{2a}\]

Step 5: Substitute Values into Formula

Substitute \(a = -3\), \(b = 11\), and \(D = 61\) into the formula:\[x = \frac{-11 \pm \sqrt{61}}{2(-3)}\]

Step 6: Calculate Each Root

Now, calculate each root:1. For the root \(x_1\):\[x_1 = \frac{-11 + \sqrt{61}}{-6}\]Using a calculator, calculate \(\sqrt{61} \approx 7.81\):\[x_1 = \frac{-11 + 7.81}{-6} = \frac{-3.19}{-6} \approx 0.53\]2. For the root \(x_2\):\[x_2 = \frac{-11 - \sqrt{61}}{-6}\]\[x_2 = \frac{-11 - 7.81}{-6} = \frac{-18.81}{-6} \approx 3.14\]

Key Concepts

Discriminant

The discriminant is a key component in solving quadratic equations. It is represented by the symbol \(D\) and is calculated using the formula:

• \(D = b^2 - 4ac\)

This calculation helps determine the nature of the roots of the quadratic equation \(ax^2 + bx + c = 0\). In this formula, \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation:

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

The discriminant reveals important information:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, often called a repeated or double root.
• If \(D < 0\), the equation has no real roots, but two complex roots.

In our specific problem, the discriminant is calculated as \(D = 61\), which indicates that there are two distinct real roots.

Real Roots

Real roots are the solutions to a quadratic equation that can be expressed as real numbers. When we talk about real roots, we are referring to the actual intersection points of the quadratic curve \(y = ax^2 + bx + c\) with the x-axis on a graph.
These points are where the value of \(y\), or the output of the quadratic equation, is zero. Real roots can be one of the following:

• Two Distinct Real Roots: This happens when the discriminant \(D > 0\). The graph touches the x-axis at two points.
• One Real Root: Occurs when \(D = 0\). The graph is tangent to the x-axis, touching it at just one point.
• No Real Roots: If \(D < 0\), the graph does not touch the x-axis at all, indicating the presence of complex roots instead of real ones.

In the exercise given, with a discriminant of 61, we know there are two distinct real roots. Calculating them provides the specific points where this quadratic crosses the x-axis.

Quadratic Formula

The quadratic formula is a universal method used to find the roots of any quadratic equation \(ax^2 + bx + c = 0\). This is how the formula appears:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula allows us to solve for \(x\) by calculating the two potential values separated by the plus-minus (\(\pm\)) sign, representing the two real roots of the equation.
The quadratic formula incorporates the discriminant \(b^2 - 4ac\) under the square root, which is pivotal in determining the nature of the roots:

• If \(D > 0\), the two calculations with \(\pm\) give two different real roots.
• If \(D = 0\), both calculations will result in the same root, reflecting the double root.
• If \(D < 0\), the square root of a negative number results in complex numbers.

For the equation \(-3x^2 + 11x - 5 = 0\) in our example, we use the quadratic formula by substituting \(a = -3\), \(b = 11\), and the discriminant value \(D = 61\).
By calculating these steps, the resulting roots are approximately \(0.53\) and \(3.14\), providing precise solutions to the equation.