Problem 69 - Algebra for College Students

Question

Use the quadratic formula and a calculator to solve each equation. Round answers to three decimal places and check your answers. $$x^{2}-7.4 x+13.69=0$$

Step-by-Step Solution

Answer

The solution rounded to three decimal places is \(x = 3.700\).

Step 1 - Identify coefficients

The given quadratic equation is in the form \(ax^2 + bx + c = 0\). Compare this with the given equation \(x^2 - 7.4x + 13.69 = 0\) to identify the coefficients: \(a = 1\), \(b = -7.4\), and \(c = 13.69\).

Step 2 - Write down the quadratic formula

The quadratic formula to find the roots of a quadratic equation is \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\].

Step 3 - Calculate the discriminant

The discriminant (\(D\)) is given by \(b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into this formula: \[D = (-7.4)^2 - 4(1)(13.69)\]. After calculating, \[D = 54.76 - 54.76 = 0\].

Step 4 - Apply the quadratic formula

Since the discriminant is 0, there is one real solution. Substitute \(a\), \(b\), and \(D\) back into the quadratic formula: \[x = \frac{{-(-7.4) \pm \sqrt{0}}}{2(1)}\]. Simplifying this, \[x = \frac{{7.4}}{2} = 3.7\].

Step 5 - Round and verify the solution

The solution to the equation rounded to three decimal places is \(x = 3.700\). Verify this by substituting \(x = 3.7\) back into the original equation: \((3.7)^2 - 7.4(3.7) + 13.69\) which equals \(13.69 - 27.38 + 13.69\) resulting in \(0\). Therefore, the solution is correct.

Key Concepts

Quadratic Equation

A quadratic equation is any equation that can be written in the form \[ax^2 + bx + c = 0,\]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This type of equation forms a parabola when graphed. In our original exercise, the quadratic equation given is \[x^2 - 7.4x + 13.69 = 0.\]Here, we can identify the coefficients as \(a = 1\), \(b = -7.4\), and \(c = 13.69\).

To solve a quadratic equation, we often use the quadratic formula, which is \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}.\]This formula allows us to find the roots or solutions of the equation by substituting the values of \(a\), \(b\), and \(c\) into the formula and simplifying.

Quadratic equations are a fundamental part of algebra and appear in various real-world scenarios, such as calculating areas, projectile motion, and economics.

Discriminant

The discriminant of a quadratic equation is a key value that helps us determine the nature and number of solutions (roots) of the equation. It is given by the expression \[D = b^2 - 4ac.\]
In the context of our exercise, the discriminant for the equation \(x^2 - 7.4x + 13.69 = 0\) was calculated as \[(-7.4)^2 - 4(1)(13.69) = 54.76 - 54.76 = 0.\]

The value of the discriminant helps us in the following ways:

• If \(D > 0\), the quadratic equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, which in this case results in a repeated root.
• If \(D < 0\), the equation has no real solutions but two complex solutions.

In our example, since the discriminant is 0, the equation has exactly one real solution.

Real Solution

A real solution of a quadratic equation is a solution that is a real number, as opposed to a complex number. Let's illustrate this using the quadratic formula. Given the equation \[x^2 - 7.4x + 13.69 = 0,\]we identified that the discriminant is 0.

According to the quadratic formula: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a},\]we substitute \(b = -7.4\), \(a = 1\), and \(D = 0\) into the formula to get: \[x = \frac{{-(-7.4) \pm \sqrt{0}}}{2(1)} = \frac{7.4}{2} = 3.7.\]

Since the discriminant is 0, there is only one real solution, which is \(x = 3.7\). This implies that the parabola touches the x-axis at just one point, indicating a double root. Real solutions are particularly important in contexts like physics, engineering, and other fields that require concrete, measurable results.