Solve each equation. (x+4)(x+2) = 2x

Verified step by step guidance

1. Start by expanding the left-hand side of the equation using the distributive property (FOIL method): \( (x+4)(x+2) = x \cdot x + x \cdot 2 + 4 \cdot x + 4 \cdot 2 \).
2. Simplify the expanded expression to get a quadratic expression: \( x^2 + 2x + 4x + 8 \).
3. Combine like terms on the left side: \( x^2 + 6x + 8 \).
4. Rewrite the equation by bringing all terms to one side to set the equation equal to zero: \( x^2 + 6x + 8 - 2x = 0 \), which simplifies to \( x^2 + 4x + 8 = 0 \).
5. Solve the quadratic equation \( x^2 + 4x + 8 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1 \), \( b=4 \), and \( c=8 \). Calculate the discriminant \( b^2 - 4ac \) first to determine the nature of the roots.

Key Concepts

Expanding Algebraic Expressions

Expanding involves multiplying out the factors in an expression to write it as a sum or difference of terms. For example, (x+4)(x+2) expands to x² + 2x + 4x + 8, which simplifies to x² + 6x + 8. This step is essential to transform the equation into a standard polynomial form.

Setting the Equation to Zero

To solve polynomial equations, it is important to rearrange all terms so that one side equals zero. This allows the use of factoring or other methods to find the roots. For the given equation, after expansion, subtract 2x from both sides to get a quadratic equation set to zero.

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula. Once the equation is in standard form ax² + bx + c = 0, find values of x that satisfy it. Factoring is often the quickest method if the quadratic factors easily.