Problem 28 - Algebra and Trigonometry

Question

Find all real solutions of the equation. \(x^{2}+30 x+200=0\)

Step-by-Step Solution

Answer

The real solutions are \( x = -10 \) and \( x = -20 \).

Step 1: Identify the Quadratic Equation

The given equation is a quadratic equation in the form of \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 30\), and \(c = 200\).

Step 2: Calculate the Discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). Substituting the values, we have \( \Delta = 30^2 - 4 \times 1 \times 200 \).

Step 3: Simplify the Discriminant

Calculate \(30^2 = 900\) and \(4 \times 1 \times 200 = 800\). Thus, \( \Delta = 900 - 800 = 100\).

Step 4: Determine the Nature of Roots

Since the discriminant \( \Delta = 100 \) is greater than 0, the quadratic equation has two distinct real solutions.

Step 5: Use the Quadratic Formula

The solutions to the quadratic equation \( ax^2 + bx + c = 0 \) are found using the quadratic formula: \( x = \frac{{-b \pm \sqrt{\Delta}}}{2a} \). Substitute \(a = 1\), \(b = 30\), and \( \Delta = 100 \).

Step 6: Substitute Values into the Quadratic Formula

The solutions are \( x = \frac{{-30 \pm \sqrt{100}}}{2 \times 1} \). Simplifying further gives \( x = \frac{{-30 \pm 10}}{2} \).

Step 7: Calculate Each Solution

Compute each solution separately. For \( x_1 = \frac{{-30 + 10}}{2} = \frac{-20}{2} = -10 \). For \( x_2 = \frac{{-30 - 10}}{2} = \frac{-40}{2} = -20 \).

Step 8: Verify the Solutions

Verify by substituting \( x = -10 \) and \( x = -20 \) back into the original equation to confirm they satisfy \( x^2 + 30x + 200 = 0 \).

Key Concepts

Discriminant

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant helps us understand the nature and number of solutions. The discriminant, denoted as \(\Delta\), is calculated using the formula \(\Delta = b^2 - 4ac\). It plays a critical role in determining whether the solutions are real or complex, and whether they are distinct or repetitive.
The value of the discriminant indicates:

• When \(\Delta > 0\), there are two distinct real solutions.
• When \(\Delta = 0\), there is exactly one real solution (often called a repeated or double root).
• When \(\Delta < 0\), there are no real solutions, only complex solutions.

In our example, with \(a = 1\), \(b = 30\), and \(c = 200\), the discriminant is calculated as \(\Delta = 30^2 - 4 \times 1 \times 200 = 100\). Since \(100 > 0\), we know there are two real and distinct solutions.

Quadratic Formula

The quadratic formula is a tool for solving quadratic equations. It is derived from the process of completing the square and is given by:\[x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a}\]This formula provides a method to find the solutions of any quadratic equation, and it can handle both real and complex roots.

Using the quadratic formula involves two main steps:

• Substitute the values of \(a\), \(b\), and \(c\) from the quadratic equation \(ax^2 + bx + c = 0\).
• Calculate the two possible solutions using the \(+\) and \(-\) in \(\pm\).

For our specific equation, substituting \(a = 1\), \(b = 30\), and the discriminant \(\sqrt{100}\) into the formula gives us the solutions \(x = \frac{{-30 \pm 10}}{2}\). Simplifying this provides the specific values of \(x = -10\) and \(x = -20\).

Real Solutions

Real solutions are the values of \(x\) that satisfy the quadratic equation and lie within the set of real numbers. They can be found using various methods, including factoring, completing the square, or using the quadratic formula.

The presence of real solutions depends on the discriminant, \(\Delta\):

• If \(\Delta > 0\), there are two real and distinct solutions.
• If \(\Delta = 0\), there is exactly one real solution.
• If \(\Delta < 0\), there are no real solutions.

In our example, since \(\Delta = 100 > 0\), we concluded that the equation has two real solutions. By using the quadratic formula, we found these solutions to be \(-10\) and \(-20\). These values satisfy the original equation as verified by substitution.

Nature of Roots

The nature of roots tells us about the characteristics of the solutions to the quadratic equation, based on the discriminant's value. Understanding this helps in predicting the behavior of the parabola represented by the quadratic equation, \(y = ax^2 + bx + c\).

This is what the discriminant reveals about the roots:

• Real and Distinct: When \(\Delta > 0\), the graph of the quadratic equation will intersect the x-axis at two distinct points.
• Real and Repeated: When \(\Delta = 0\), the parabola is tangent to the x-axis, touching it at exactly one point.
• Complex Roots: When \(\Delta < 0\), the parabola does not intersect the x-axis, indicating that there are no real x-intercepts.

Therefore, for the equation \(x^2 + 30x + 200 = 0\), we see two real and distinct roots because \(\Delta = 100 > 0\). This is reflected in the solutions \(-10\) and \(-20\).