Problem 111 - Algebra for College Students

Question

For each equation, find approximate solutions rounded to two decimal places. $$x^{2}-7.3 x+12.5=0$$

Step-by-Step Solution

Answer

The approximate solutions are x = 4.56 and x = 2.75.

Step 1: Identify coefficients

The quadratic equation is in the form of ax^{2} + bx + c = 0. Identify the coefficients a, b, and c.Here, a = 1, b = -7.3, and c = 12.5.

Step 2: Apply the quadratic formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).Plug in the identified coefficients:\(a = 1\), \(b = -7.3\), and \(c = 12.5\).

Step 3: Calculate the discriminant

First, calculate the discriminant:\(b^{2} - 4ac\).\((-7.3)^{2} - 4 \times 1 \times 12.5\).\(53.29 - 50 = 3.29\).

Step 4: Calculate the roots

Now, find the two solutions using the quadratic formula:\(x = \frac{-(-7.3) \pm \sqrt{3.29}}{2 \times 1}\).\(x = \frac{7.3 \pm 1.81}{2}\).

Step 5: Find the approximate solutions

Solve for the two roots:\(x_{1} = \frac{7.3 + 1.81}{2} = \frac{9.11}{2} = 4.56\).\(x_{2} = \frac{7.3 - 1.81}{2} = \frac{5.49}{2} = 2.75\).

Step 6: Round off the solutions

The approximate solutions, rounded to two decimal places, are\(x_{1} \approx 4.56\) and \(x_{2} \approx 2.75\).

Key Concepts

Quadratic Formula

The quadratic formula is essential for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The formula allows us to find the roots of the quadratic equation by using the coefficients \(a\), \(b\), and \(c\). The quadratic formula is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

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

To apply the quadratic formula, you simply plug in the values of \(a\), \(b\), and \(c\) from your quadratic equation into the formula. Next, you will need to calculate the discriminant, which we will talk about in the following section.

Discriminant

The discriminant is the part of the quadratic formula under the square root:

\(b^2 - 4ac\).

• If the discriminant is positive, there are two real and distinct roots.
• If it is zero, there is exactly one real root (also called a repeated root).
• If the discriminant is negative, there are no real roots, but two complex roots.

In our example equation \(x^2 - 7.3x + 12.5\), the discriminant is calculated as follows:

\[ b^2 - 4ac = (-7.3)^2 - 4(1)(12.5) = 53.29 - 50 = 3.29 \]

Since the discriminant is positive \( (3.29) \), our quadratic equation has two real and distinct roots.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \(x\) that make the equation true. You can think of the roots as the points where the graph of the quadratic equation intersects the x-axis. Using the quadratic formula, we can find these roots:

In our example, we found:

\[ x = \frac{-(-7.3) \pm \sqrt{3.29}}{2(1)} = \frac{7.3 \pm 1.81}{2} \]

This results in two roots:

• \(x_1 = \frac{7.3 + 1.81}{2} = 4.56\)
• \(x_2 = \frac{7.3 - 1.81}{2} = 2.75\)

These roots, \(4.56\) and \(2.75\), are where the quadratic equation \(x^2 - 7.3x + 12.5 = 0\) intersects the x-axis.