Problem 25 - Elementary Algebra

Question

\(h^{2}+h-5=0\)

Step-by-Step Solution

Answer

The solutions are \(h = \frac{-1 + \sqrt{21}}{2}\) and \(h = \frac{-1 - \sqrt{21}}{2}\).

Step 1 - Identify the equation type

The equation given is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 1\), and \(c = -5\).

Step 2 - Use the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Identify the coefficients from the equation, which are \(a = 1\), \(b = 1\), and \(c = -5\).

Step 3 - Substitute the coefficients into the formula

Substitute \(a = 1\), \(b = 1\), and \(c = -5\) into the quadratic formula: \[h = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-5)}}{2(1)}\].

Step 4 - Simplify under the square root

Calculate the value under the square root: \((1)^2 - 4(1)(-5) = 1 + 20 = 21\).

Step 5 - Solve the equation

Now the equation becomes \[h = \frac{-1 \pm \sqrt{21}}{2}\]. Split this into two solutions: \(h_1 = \frac{-1 + \sqrt{21}}{2}\) and \(h_2 = \frac{-1 - \sqrt{21}}{2}\).

Key Concepts

Quadratic Formula

One effective method to solve quadratic equations is through the quadratic formula.
This formula allows you to find the roots of a quadratic equation without factoring or graphing.
The quadratic formula is provided as: ewline ewline \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
where \(a\), \(b\), and \(c\) are coefficients from the standard form \(ax^2 + bx + c = 0\).
The term inside the square root, \(b^2 - 4ac\), is called the discriminant.
It determines the nature of the roots.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation.
It is often written in the standard form: \(ax^2 + bx + c = 0\), where \(a e 0\).

Characteristics of a quadratic equation:

• It contains a variable that is squared (\(x^2\)).
• The highest degree of the variable is two.
• It can have at most two solutions, called roots.

In our exercise \(h^2 + h - 5 = 0\), the coefficients are \(a = 1\), \(b = 1\), and \(c = -5\).
These coefficients are crucial for solving the equation using the quadratic formula.

Roots of Equation

The roots of a quadratic equation are the values of the variable that satisfy the equation.
They are also known as solutions or zeros of the equation.
In our example \(h^2 + h - 5 = 0\), the roots can be found using the quadratic formula.

By substituting the coefficients \(a = 1\), \(b = 1\), and \(c = -5\) into the quadratic formula, we get:

\[h = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-5)}}{2(1)}\]

After simplifying under the square root, the discriminant becomes \(21\), leading us to:

\[h = \frac{-1 \pm \sqrt{21}}{2}\]
The two roots of the equation are:

\[h_1 = \frac{-1 + \sqrt{21}}{2}\] and \[h_2 = \frac{-1 - \sqrt{21}}{2}\]

Solving Step-by-Step

Understanding each step in solving a quadratic equation ensures you grasp the full concept.
Let’s break down the solution:

• Step 1: Identify the quadratic equation \(h^2 + h - 5 = 0\).
• Step 2: Note the coefficients \(a = 1\), \(b = 1\), and \(c = -5\).
• Step 3: Use these coefficients in the quadratic formula: \[h = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-5)}}{2(1)}\]
• Step 4: Simplify the discriminant to get \(1 + 20 = 21\).
• Step 5: Solve for \(h\) resulting in \(h = \frac{-1 \pm \sqrt{21}}{2}\).
• Step 6: Split this into our two roots: \(h_1 = \frac{-1 + \sqrt{21}}{2}\) and \(h_2 = \frac{-1 - \sqrt{21}}{2}\).

Each step is crucial to wholly understand how the quadratic formula works.
It also demystifies the method by breaking it down into smaller, more manageable parts.