Problem 9 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$x^{2}+5 x+6=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -2\) and \(x = -3\).

Step 1 - Identify coefficients

In the quadratic equation, identify the coefficients. For the equation \(x^2 + 5x + 6 = 0\), the coefficients are: \(a = 1\), \(b = 5\), and \(c = 6\).

Step 2 - Write Quadratic Formula

Recall the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3 - Substitute coefficients

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} \]

Step 4 - Simplify under the square root (discriminant)

Calculate inside the square root: \[ 5^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1 \] So, the equation now is: \[ x = \frac{-5 \pm \sqrt{1}}{2} \]

Step 5 - Simplify further

Since \(\sqrt{1} = 1\), the equation becomes: \[ x = \frac{-5 \pm 1}{2} \]

Step 6 - Find solutions

Split into two solutions: \[ x_1 = \frac{-5 + 1}{2} = \frac{-4}{2} = -2 \] \[ x_2 = \frac{-5 - 1}{2} = \frac{-6}{2} = -3 \]

Step 7 - State final answers

The solutions to the equation are \(x = -2\) and \(x = -3\).

Key Concepts

Solving Quadratic Equations

To solve quadratic equations, one effective method is using the quadratic formula. Quadratic equations are any equations of the form \(ax^2 + bx + c = 0\). In this form, \(a, b,\) and \(c\) are constants, with \(a \e 0\). The quadratic formula derives from the process of completing the square, and it provides a general solution to all quadratic equations. Here’s the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] By following specific steps, you can find the roots (solutions) of any quadratic equation. These steps include identifying coefficients, substituting them into the formula, simplifying the expression under the square root, and then finding the final values of \(x\). This method is reliable and works for all standard quadratic equations.

Discriminant

The discriminant is a key part of the quadratic formula and is found under the square root symbol. It is expressed with the formula \(b^2 - 4ac\). This value can tell you about the nature of the roots of the quadratic equation:

• If the discriminant is positive (greater than zero), there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root (a repeated root).
• If the discriminant is negative (less than zero), there are no real roots, but two complex (imaginary) roots instead.

Considering the discriminant helps understand whether to expect real or complex solutions. For example, in the equation given, \(x^2 + 5x + 6 = 0\), the discriminant is calculated as \(25 - 24 = 1\), which is positive, indicating two distinct real roots.

Quadratic Equation Solutions

Using the quadratic formula to find quadratic equation solutions involves several steps. After calculating the discriminant and taking its square root, you use it to solve for the roots \(x\). Given that \( \sqrt{1} = 1 \), the full equation now reads: \[ x = \frac{-5 \pm 1}{2} \] This results in two values:

• For the plus sign: \(x_1 = \frac{-5 + 1}{2} = \frac{-4}{2} = -2 \)
• For the minus sign: \(x_2 = \frac{-5 - 1}{2} = \frac{-6}{2} = -3 \)

These are the solutions to the quadratic equation. When interpreting these results, it's vital to understand they represent the points where the quadratic function intersects the x-axis in a Cartesian plane.

Coefficients in Quadratic Equations

In any quadratic equation \(ax^2 + bx + c = 0\), the coefficients \(a, b,\) and \(c\) play crucial roles in determining the equation’s solutions.

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