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

Verified step by step guidance

1. Start by expanding the left-hand side of the equation using the distributive property (FOIL method): multiply each term in the first binomial by each term in the second binomial. So, expand \((2x+1)(x-4)\) to get \(2x \cdot x + 2x \cdot (-4) + 1 \cdot x + 1 \cdot (-4)\).
2. Simplify the expression from the expansion: combine like terms to write the left side as a quadratic expression in standard form.
3. Rewrite the equation by setting it equal to the right-hand side, which is \(x\). Then, move all terms to one side of the equation to set the equation equal to zero. This will give you a quadratic equation in the form \(ax^2 + bx + c = 0\).
4. Use the quadratic formula to solve for \(x\). Recall the quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients from your quadratic equation.
5. Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots, then substitute the values into the quadratic formula to find the solutions for \(x\).

Key Concepts

Expanding Algebraic Expressions

Expanding involves multiplying terms within parentheses to remove them, using the distributive property. For example, (2x+1)(x-4) expands to 2x·x + 2x·(-4) + 1·x + 1·(-4). This step simplifies the equation into a polynomial form that is easier to solve.

Forming and Solving Quadratic Equations

After expansion, the equation typically becomes quadratic (ax² + bx + c = 0). Solving quadratic equations involves rearranging all terms to one side to set the equation equal to zero, then applying methods like factoring, completing the square, or the quadratic formula to find the values of x.

Isolating Variables and Simplifying Equations

Isolating variables means manipulating the equation to get x alone on one side. This includes combining like terms and moving all terms to one side to simplify the equation. Simplification is crucial before applying solution methods to ensure accuracy and clarity.