Problem 50 - Algebra 1: Concepts and Skills

Question

Write the quadratic equation in standard form. Then solve using the quadratic formula. $$x^{2}+3 x=-2$$

Step-by-Step Solution

Answer

The solutions for the given quadratic equation are \(x = -1\) and \(x = -2\).

Step 1: Transform to Standard Form

Rearrange the equation given, \(x^2 + 3x = -2\), to match the standard form. This can be done by transferring -2 from the right-hand side (RHS) to the left-hand side (LHS). Evidence of this process is as shown below:\n\(x^2 + 3x + 2 = 0\). Here, the quadratic equation is in its standard form, where \(a = 1\), \(b = 3\) and \(c = 2\).

Step 2: Implement the Quadratic Formula

Plug these values into the quadratic equation which is given by \(-b \pm \sqrt{b^2 - 4ac} / 2a\). Thus the solution will be \(-b \pm \sqrt{b^2 -4ac} / 2a = -(3) \pm \sqrt{(3)^2 -4*(1)*(2)} / 2*(1) = -3 \pm \sqrt{9 -8} / 2 = -3 \pm \sqrt{1} / 2\). Therefore, after calculating the square root we get, \(-3 \pm 1 / 2\).

Step 3: Find the Solution

The solutions are given by the following equation: \(-3 \pm 1 / 2 = x\). So, there are two solutions. 1) If the plus sign is considered in the equation, then \(x = (-3+1) / 2 = -1\) 2) If the minus sign is considered, then \(x = (-3-1) / 2 = -2\). Therefore, the solutions to the equation \(x^2 +3x + 2 = 0\) are \(x=-1\) and \(x=-2\).

Key Concepts

Standard Form

When dealing with quadratic equations, it is essential to know the concept of "Standard Form." A quadratic equation is typically a polynomial equation of degree 2. It is expressed as:

• \( ax^2 + bx + c = 0 \)

In this format:

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

To transform a given quadratic equation into its standard form, rearrange all terms to one side so that the equation equals zero.
In our example, starting with \( x^2 + 3x = -2 \), we move \( -2 \) to the left side:

• \( x^2 + 3x + 2 = 0 \)

Now, the quadratic equation is in standard form with \( a = 1 \), \( b = 3 \), and \( c = 2 \). This uniform format is crucial for using various methods to solve quadratic equations, including the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions of quadratic equations. It applies to any quadratic equation in standard form:

• \( ax^2 + bx + c = 0 \)

The quadratic formula is expressed as:

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

This formula derives from completing the square and offers a universal method to solve any quadratic equation. Let's break it down:

• \( b^2 - 4ac \) is known as the "discriminant." It determines the number and type of solutions:
• If it is positive, there are two distinct real solutions.
• If it is zero, there's exactly one real solution.
• If it's negative, the equation has two complex solutions.

For our example \( x^2 + 3x + 2 = 0 \):

• Calculate \( b^2 - 4ac \): \( 3^2 - 4 \times 1 \times 2 \)
• This equals \( 9 - 8 = 1 \), a positive value indicating two real solutions.

Plug these values into the quadratic formula to find \( x \).

Solutions of Quadratic Equations

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation. After substituting into the quadratic formula, we can find the solutions.
For our example, we substitute the coefficients \( a = 1 \), \( b = 3 \), and \( c = 2 \) into the quadratic formula:

• \( x = \frac{-3 \pm \sqrt{1}}{2 \times 1} \)
• So, \( x = \frac{-3 \pm 1}{2} \)

The "plus-minus" symbol \( \pm \) indicates two calculations:

• \( x = \frac{-3 + 1}{2} = -1 \)
• \( x = \frac{-3 - 1}{2} = -2 \)

Thus, the solutions are \( x = -1 \) and \( x = -2 \). These are the points where the quadratic equation intersects the x-axis on a graph. Solving using these methods gives a clear view of where solutions lie and how they relate to the equation's structure.