Problem 6 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$x^{2}+20 x=25$$

Step-by-Step Solution

Answer

The solutions are \( x = -10 + 5\sqrt{5} \) and \( x = -10 - 5\sqrt{5} \).

Step 1: Rearrange the Equation

Rearrange the given quadratic equation into the standard form: \[ x^2 + 20x - 25 = 0 \]

Step 2: Identify Values for the Quadratic Formula

For the quadratic formula, the equation \( ax^2 + bx + c = 0 \) requires identifying \( a \), \( b \), and \( c \). Here, \( a = 1 \), \( b = 20 \), and \( c = -25 \).

Step 3: Set Up the Quadratic Formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute \( a = 1 \), \( b = 20 \), and \( c = -25 \) into the formula.

Step 4: Calculate the Discriminant

Compute the discriminant \( b^2 - 4ac \):\[ 20^2 - 4 \times 1 \times (-25) = 400 + 100 = 500 \]

Step 5: Solve the Quadratic Equation

Since the discriminant is 500, substitute it back into the formula: \[ x = \frac{-20 \pm \sqrt{500}}{2} \]Calculate \( \sqrt{500} = 10\sqrt{5} \), thus the solutions are:\[ x = \frac{-20 \pm 10\sqrt{5}}{2} \]

Step 6: Simplify the Solutions

Simplify each component of the expression:\[ x = \frac{-20 \pm 10\sqrt{5}}{2} = -10 \pm 5\sqrt{5} \]Thus, the solutions are \( x = -10 + 5\sqrt{5} \) and \( x = -10 - 5\sqrt{5} \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It provides a reliable way to find the roots or solutions of the equation, even when other methods might be too complex or not possible. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The variables \( a \), \( b \), and \( c \) are coefficients from the quadratic equation. By substituting these values into the formula, you can calculate the roots.

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

Using the quadratic formula not only gives you the numerical solutions, but also helps understand the nature of the roots, whether they are real or complex.

Discriminant

The discriminant is a key part of the quadratic formula and can tell us much about the nature of the roots of the quadratic equation. It is found under the square root sign in the quadratic formula:\[d = b^2 - 4ac\]
The value of the discriminant determines the type of solutions the quadratic equation will have:

• If the discriminant \( d > 0 \), the quadratic equation has two distinct real roots.
• If \( d = 0 \), there is one real root, which means the parabola touches the x-axis at a single point.
• If \( d < 0 \), the equation has two complex roots, indicating the parabola does not intersect the x-axis.

This understanding helps in predicting the behavior of the solutions even before you calculate them.

Solving Equations

Solving quadratic equations is all about finding the values of \( x \) that satisfy the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. When the coefficients do not easily lend themselves to factoring, the quadratic formula is particularly useful.
When applying the quadratic formula:

• First, ensure your equation is in standard form, \( ax^2 + bx + c = 0 \).
• Identify the coefficients \( a \), \( b \), and \( c \).
• Calculate the discriminant \( b^2 - 4ac \) to anticipate the nature of the solutions.
• Substitute into the quadratic formula and simplify to find the roots.

This structured approach ensures accurate solutions and offers insight into the mathematics behind the quadratic equation, strengthening overall problem-solving skills.