Problem 38 - Elementary Algebra

Question

\(x^{2}+7 x-9=0\)

Step-by-Step Solution

Answer

The solutions are \( x = \frac{-7 + \sqrt{85}}{2} \) and \( x = \frac{-7 - \sqrt{85}}{2} \).

Step 1: Identify the coefficients

In the quadratic equation of the form \( ax^2 + bx + c = 0 \), identify the coefficients. For the equation \( x^2 + 7x - 9 = 0 \), \( a=1 \), \( b=7 \), and \( c=-9 \).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation is given by \( \Delta = b^2 - 4ac \). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \( \Delta = 7^2 - 4(1)(-9) \). Simplify to get \( \Delta = 49 + 36 = 85 \).

Step 3: Apply the quadratic formula

The solutions to the quadratic equation can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \). Substitute \( a=1 \), \( b=7 \), and \( \Delta=85 \) into the formula to get \( x = \frac{-7 \pm \sqrt{85}}{2} \).

Step 4: Simplify the solutions

Separate the two solutions using the plus and minus signs: \( x_1 = \frac{-7 + \sqrt{85}}{2} \) and \( x_2 = \frac{-7 - \sqrt{85}}{2} \).

Key Concepts

Quadratic Formula

To solve any quadratic equation, especially those that don't factor easily, we use the quadratic formula. The quadratic formula is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this formula:

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

This formula works for any quadratic equation of the form \( ax^2 + bx + c = 0 \). It's very useful because it provides a straightforward way to find the roots of the equation directly. Additionally, regardless of the complexity of \( a \), \( b \), and \( c \), the quadratic formula can solve for \( x \).

Discriminant

The discriminant is a key part of the quadratic formula, found under the square root in the formula:

\[ \sqrt{b^2 - 4ac} \]

This value \( b^2 - 4ac \) is known as the discriminant \( \Delta \). The discriminant tells us a lot about the nature of the roots of the quadratic equation.

• If \( \Delta > 0 \), we have two distinct real roots.
• If \( \Delta = 0 \), we have exactly one real root (a repeated root).
• If \( \Delta < 0 \), we have two complex roots (not real).

For the equation \( x^2 + 7x - 9 = 0 \), the discriminant is calculated as:

\[ \Delta = 7^2 - 4(1)(-9) = 49 + 36 = 85 \]

Since \( \Delta = 85 \) which is greater than 0, this means we have two distinct real roots.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \( x \) that make the equation true. You can solve these equations using different methods:

• Factoring (if the equation can be factored easily)
• Using the quadratic formula
• Completing the square

In our example, we use the quadratic formula because factoring \( x^2 + 7x - 9 = 0 \) might not be straightforward.

Using the quadratic formula:

\[ x = \frac{-7 \pm \sqrt{85}}{2} \]

Breaking it down, we get the solutions:\[ x_1 = \frac{-7 + \sqrt{85}}{2} \] and \[ x_2 = \frac{-7 - \sqrt{85}}{2} \]

These two values of \( x \) are the roots of our quadratic equation.

Coefficients in Quadratics

Coefficients play a critical role in quadratic equations. These are the numerical factors that multiply the variable terms in the equation:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term (or the coefficient of \( x^0 \))

In the given equation \( x^2 + 7x - 9 = 0 \), we identify the coefficients as:

• \( a = 1 \) (since there is 1 \( x^2 \) term)
• \( b = 7 \) (the coefficient of the \( x \) term)
• \( c = -9 \) (the constant term term)

Accurately identifying these coefficients is the first step in correctly applying the quadratic formula and solving the equation.