Problem 20 - Algebra A Combined Function

Question

Solve. $$ x^{2}+2 x-63=0 $$

Step-by-Step Solution

Answer

The roots are \( x = 7 \) and \( x = -9 \).

Step 1: Identify the Equation Type

The given equation is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = 2 \), and \( c = -63 \).

Step 2: Compute the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \). Substitute the values: \( (2)^2 - 4(1)(-63) = 4 + 252 = 256 \).

Step 3: Check the Nature of the Roots

Since the discriminant is positive (256), the quadratic equation has two distinct real roots.

Step 4: Use the Quadratic Formula

The roots are given by the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the known values into the formula: \( x = \frac{-2 \pm \sqrt{256}}{2(1)} = \frac{-2 \pm 16}{2} \).

Step 5: Calculate the Roots

Evaluate the two possible solutions:1. \( x = \frac{-2 + 16}{2} = 7 \).2. \( x = \frac{-2 - 16}{2} = -9 \).Thus, the roots are \( x = 7 \) and \( x = -9 \).

Key Concepts

Discriminant

The discriminant is a key part of solving quadratic equations. It reveals much about the equation's roots, which are the solutions you're seeking. Given a standard quadratic equation in the form of \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) can be calculated using the formula:

• \( \Delta = b^2 - 4ac \)

In our problem, with \( a = 1 \), \( b = 2 \), and \( c = -63 \), the discriminant is calculated as:

• \( (2)^2 - 4(1)(-63) = 4 + 252 = 256 \)

A positive discriminant (like our \( 256 \)) indicates that the quadratic equation will have two distinct real roots. If it's zero, expect a repeated real root, and if it's negative, the roots will be complex or non-real. Hence, knowing the discriminant isn't just about solving—it’s about understanding the nature of the roots you'll find.

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions to any quadratic equation. It's easy to use. Just plug in the values and watch it work. The formula is:

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

You use this formula when the quadratic equation is in its standard form, \( ax^2 + bx + c = 0 \). In our exercise, substituting \( a = 1 \), \( b = 2 \), and \( c = -63 \) into the formula, we get:

• \( x = \frac{-2 \pm \sqrt{256}}{2(1)} \)

Which simplifies to:

• \( x = \frac{-2 \pm 16}{2} \)

So, we get two roots:

• \( x = \frac{-2 + 16}{2} = 7 \)
• \( x = \frac{-2 - 16}{2} = -9 \)

These are the solutions to the equation. This formula will always guide you to the roots, whether they're real or complex.

Nature of Roots

Understanding the nature of roots before solving helps set your expectations. It involves knowing whether roots are real, distinct, repeated, or even complex. This is directly informed by the discriminant we discussed earlier. Let's use our recent discriminant insight for this:

• If \( \Delta > 0 \), the equation has two distinct real roots (like in our equation).
• If \( \Delta = 0 \), expect exactly one real root repeated twice.
• If \( \Delta < 0 \), the roots will be complex and not visible on the real number line.

For the quadratic equation \( x^2 + 2x - 63 = 0 \), the discriminant (256) confirms our roots are distinct and real. Hence, by calculating roots using the quadratic formula, you can confidently know what type of solutions to expect from the start. This step saves time and helps anticipate what results to look for.