Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}+5 x+4=0 $$

Step-by-Step Solution

Answer

The roots are \(x = -1\) and \(x = -4\).

Step 1: Identify a, b, and c

In the given quadratic equation \(x^2 + 5x + 4 = 0\), we identify the coefficients: \(a = 1\), \(b = 5\), and \(c = 4\). These values will be used in the quadratic formula.

Step 2: Apply the Quadratic Formula

Use the quadratic formula, which is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b = 5\), and \(c = 4\) into the formula.

Step 3: Calculate the Discriminant

The discriminant is the part of the quadratic formula under the square root: \(b^2 - 4ac\). Calculate it as follows: \((5)^2 - 4(1)(4) = 25 - 16 = 9\).

Step 4: Calculate the Roots

Since the discriminant is 9, a perfect square, the square root simplifies to 3. Substitute back into the formula: \(x = \frac{-5 \pm 3}{2(1)}\). This gives two solutions: \(x = \frac{-5 + 3}{2}\) and \(x = \frac{-5 - 3}{2}\).

Step 5: Simplify the Roots

Simplify the solutions from Step 4: \(x = \frac{-2}{2} = -1\) and \(x = \frac{-8}{2} = -4\). Both roots are integers.

Key Concepts

Discriminant

The discriminant is a significant component in solving quadratic equations using the quadratic formula. Located under the square root sign in the formula, it is calculated as \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of a quadratic equation.
To find the discriminant, you simply plug in the coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(ax^2 + bx + c = 0\). Let's take the example from the exercise: for the equation \(x^2 + 5x + 4 = 0\), \(a = 1\), \(b = 5\), and \(c = 4\). Thus, the discriminant \(D\) is calculated as:

• \(D = b^2 - 4ac\)
• \(D = (5)^2 - 4(1)(4)\)
• \(D = 25 - 16\)
• \(D = 9\)

The value of the discriminant tells us about the roots:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root (or two identical real roots).
• If \(D < 0\), the roots are complex and not real.

In our case, since the discriminant is 9 (a positive perfect square), we will have two distinct real roots.

Quadratic Equation

Quadratic equations are polynomial equations of degree 2 and have the general form \(ax^2 + bx + c = 0\). They are called 'quadratic' because 'quad' means square, indicating the equation involves at least one variable squared.
The simplest quadratic equation would be of the form \(x^2 = 0\), but in practice, the equation includes linear and constant terms as well. Here's how the parts of the equation \(x^2 + 5x + 4 = 0\) function:

• \(a\), \(b\), \(c\) are constants, wherein \(aeq 0\). If \(a\) were zero, it would not be a quadratic equation.
• \(x\) represents the variable or the unknown we are trying to solve for.

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a powerful tool to find solutions (or roots) of such quadratic equations.
Every quadratic equation can potentially have two solutions, given by the two signs \(\pm\) in the formula, which correspond to the two possible values that \(x\) could take to satisfy the equation.

Roots of Equation

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). They are also known as solutions or zeros of the equation. Knowing the roots gives us vital insights into the curve represented by the quadratic equation, as they tell us where the curve intersects the x-axis.
Using the quadratic formula, the roots are found by substituting the coefficients into the formula and simplifying it:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Plug in values for our specific equation: \(x = \frac{-5 \pm \sqrt{9}}{2}\)
• Calculate the two possible values: \(x = \frac{-5 + 3}{2}\) and \(x = \frac{-5 - 3}{2}\)
• Simplify to get the roots: \(x = -1\) and \(x = -4\)

These roots tell us the points \((-1,0)\) and \((-4,0)\) where this particular quadratic curve reaches the x-axis. Finding and understanding these roots can be fundamental in graphing quadratic functions, solving for x in various problems, or even optimizing situations represented by quadratic curves.