Problem 69 - Elementary Algebra

Question

\(x^{2}+4 x=4\)

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Equation in Standard Form

First, rewrite the given equation as a standard quadratic equation of the form \(ax^2 + bx + c = 0\). Move all the terms to one side of the equation:\[ x^2 + 4x - 4 = 0 \]

Step 2: Identify Coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(ax^2 + bx + c = 0\):\(a = 1\), \(b = 4\), and \(c = -4\)

Step 3: Use the Quadratic Formula

To find the solutions for \(x\), use the quadratic formula:\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]Substitute \(a\), \(b\), and \(c\) into the formula: \[ x = \frac{{-4 \pm \sqrt{{4^2 - 4(1)(-4)}}}}{2(1)} \]

Step 4: Simplify the Expression

Simplify the expression under the square root (the discriminant) and then the entire formula: \[ x = \frac{{-4 \pm \sqrt{{16 + 16}}}}{2} \] \[ x = \frac{{-4 \pm \sqrt{32}}}{2} \] \[ x = \frac{{-4 \pm 4\sqrt{2}}}{2} \] \[ x = -2 \pm 2\sqrt{2} \]

Step 5: Write Down the Solutions

Therefore, the solutions to the equation \(x^2 + 4x = 4\) are: \[ x = -2 + 2\sqrt{2} \] and \[ x = -2 - 2\sqrt{2} \]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool that helps to solve any quadratic equation. Let's recall what a quadratic equation is: it is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula provides the solutions for \(x\) directly by substituting these constants into a single equation:

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

This formula is derived from completing the square on the standard quadratic equation. It includes a crucial component known as the discriminant (\(b^2 - 4ac\)), which determines the nature of the roots. We will talk more about the discriminant in its own section.

Standard Form of Quadratic Equation

Before applying the quadratic formula, you need to ensure that your equation is in the standard form \(ax^2 + bx + c = 0\). In the given exercise, the equation is \(x^2 + 4x = 4\).

To convert this into the standard form, we move all terms to one side of the equation:

• \(x^2 + 4x - 4 = 0\)

Here, we can identify the coefficients of the equation:

• \(a = 1\)
• \(b = 4\)
• \(c = -4\)

Now that we have these coefficients, we can easily apply the quadratic formula to find the solutions for \(x\).

Discriminant

The discriminant is a part of the quadratic formula and is represented by \(b^2 - 4ac\). It plays a key role in determining the nature of the roots of the quadratic equation:

• If the discriminant is positive (\(>0\)), the equation has two distinct real roots.
• If the discriminant is zero (\(=0\)), the equation has exactly one real root (a repeated root).
• If the discriminant is negative (\(<0\)), the equation has two complex roots (no real roots).

In our given exercise:

• \(x^2 + 4x - 4 = 0\)
• \(b = 4\)
• \(a = 1\)
• \(c = -4\)

Substitute these values into the discriminant formula:

• \(b^2 - 4ac = 4^2 - 4(1)(-4)\)
• \(= 16 + 16\)
• \(= 32\)

Since the discriminant is positive (\(32\)), the quadratic equation has two distinct real roots.