Solve each equation using the quadratic formula. See Examples 5 and 6.
$(4x-1)(x+2)=4x$

Verified step by step guidance

1. First, expand the left side of the equation by using the distributive property: multiply each term in (4x - 1) by each term in (x + 2). This gives you \(4x \cdot x + 4x \cdot 2 - 1 \cdot x - 1 \cdot 2\).
2. Simplify the expression from step 1 to get a quadratic expression in standard form. Combine like terms to write the equation as \(ax^2 + bx + c = 4x\).
3. Next, move all terms to one side of the equation to set it equal to zero. Subtract \$4x\( from both sides to get \)ax^2 + bx + c - 4x = 0$, then combine like terms again.
4. Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation in standard form \(ax^2 + bx + c = 0\). These coefficients will be used in the quadratic formula.
5. Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) using the values of \(a\), \(b\), and \(c\) found in step 4. This will give you the solutions for \(x\).

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. Understanding how to rewrite equations into this standard form is essential before applying methods like the quadratic formula.

Expanding and Simplifying Expressions

To solve the given equation, you must first expand the product (4x - 1)(x + 2) and then simplify the resulting expression. This step helps in rearranging the equation into the standard quadratic form.

Quadratic Formula

The quadratic formula x = [-b ± √(b² - 4ac)] / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It requires identifying coefficients a, b, and c correctly and calculating the discriminant to find real or complex roots.