Problem 13 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$x^{2}-2 x-10=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 1+ \sqrt{11}\) and \(x = 1 - \sqrt{11}\)

Step 1: Identify a, b, and c

From the given quadratic equation \(x^{2}-2 x-10=0\) , identify the values of \(a\), \(b\), and \(c\) as follows: \(a = 1\), \(b = -2\), \(c = -10\)

Step 2: Substitute in the Quadratic Formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-2) \pm \sqrt{(-2)^{2} - 4(1)(-10)}}{2(1)}\)

Step 3: Calculate the Discriminant

Calculate the value under the square root, known as the discriminant: \((-2)^{2} - 4(1)(-10) = 4 + 40 = 44\)

Step 4: Apply the Square Root and Simplify

Apply the square root to the discriminant and simplify the equation: \(x = \frac{2 \pm \sqrt{44}}{2}\). Now divide each term in the numerator by 2 to get \(x = 1 \pm \sqrt{11}\)

Step 5: Final Answer

So the solutions to the equation \(x^{2}-2 x-10=0\) are \(x = 1+ \sqrt{11}\) and \(x = 1 - \sqrt{11}\)

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. To solve these equations, one widely used approach is the Quadratic Formula, which is a universal formula for finding the roots (solutions) of quadratic equations.

You can use the formula:

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

Here, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation, and \( x \) represents the variable we aim to solve for. Plug these numbers into the formula, and you'll yield the solutions for \( x \).

Remember:

• \( b^2 - 4ac \) under the square root sign is called the discriminant, and it plays a crucial role in determining the nature of the roots.
• The \( \pm \) symbol indicates that there are usually two solutions.

By substituting the identified values for \( a \), \( b \), and \( c \), you can solve for \( x \) effectively.

Discriminant

The discriminant is a key part of the quadratic formula, represented by \( b^2 - 4ac \). It determines the nature of the roots for the quadratic equation. By evaluating the discriminant's value, you can ascertain characteristics about the solutions:

• If the discriminant is positive, two distinct real roots exist.
• If the discriminant is zero, both roots are real and equal, which means there's exactly one solution.
• If the discriminant is negative, there are no real solutions, but two complex ones.

For example, when solving the equation \( x^2 - 2x - 10 = 0 \), the discriminant is calculated as follows:

• Using \( a = 1 \), \( b = -2 \), and \( c = -10 \), the discriminant formula gives \( (-2)^2 - 4(1)(-10) = 4 + 40 = 44 \).
• Since 44 is positive, it signifies that there are two distinct real roots.

Understanding the discriminant aids in predicting the type of solutions you'll encounter, thus providing a more comprehensive understanding of the quadratic equation.

Simplifying Square Roots

Simplifying square roots can often clarify the final solution to a quadratic equation. After applying the quadratic formula, you might end up with a square root that isn't immediately simple. Breaking down the square root involves finding its prime factors or simplifying it to its simplest radical form.

Here's how to simplify a square root such as \( \sqrt{44} \):

• Express 44 as a product of prime factors: \( 44 = 4 \times 11 \).
• Since \( \sqrt{4} = 2 \), you can simplify \( \sqrt{44} \) to \( 2\sqrt{11} \).
• Thus, the expression \( \frac{2 \pm \sqrt{44}}{2} \) simplifies to \( \frac{2 \pm 2\sqrt{11}}{2} \), which breaks down further to \( 1 \pm \sqrt{11} \) after dividing each term by 2.

By simplifying square roots, you can present a cleaner and more precise solution, such as obtaining roots \( x = 1 + \sqrt{11} \) and \( x = 1 - \sqrt{11} \) for the equation \( x^2 - 2x - 10 = 0 \). This step enhances clarity and understanding of the solutions.