Solve each equation. (2x+1)(x-4) = x

Verified step by step guidance

1. Expand the left side of the equation: $(2x+1)(x-4)$ to get $2x(x-4)+1(x-4)$.
2. Simplify the expression: $2x^2-8x+x-4$.
3. Combine like terms: $2x^2-7x-4$.
4. Set the equation to zero by subtracting $x$ from both sides: $2x^2-7x-4-x=0$.
5. Combine like terms again: $2x^2-8x-4=0$.

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the context of the given equation, recognizing that the left side can be expanded helps in simplifying the equation for easier solving.

Distributive Property

The Distributive Property states that a(b + c) = ab + ac. This property is essential for expanding expressions like (2x + 1)(x - 4) into a polynomial form, allowing us to combine like terms and set the equation to zero for solving.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0. After expanding and rearranging the given equation, it may result in a quadratic form, which can be solved using methods such as factoring, completing the square, or the quadratic formula.