Solve each equation using completing the square. x² - 7x + 12 = 0

Verified step by step guidance

1. Start with the given quadratic equation: \(x^2 - 7x + 12 = 0\).
2. Move the constant term to the other side to isolate the quadratic and linear terms: \(x^2 - 7x = -12\).
3. To complete the square, take half of the coefficient of \(x\), which is \(-7\), divide by 2 to get \(-\frac{7}{2}\), then square it to get \(\left(-\frac{7}{2}\right)^2 = \frac{49}{4}\).
4. Add \(\frac{49}{4}\) to both sides of the equation to maintain equality: \(x^2 - 7x + \frac{49}{4} = -12 + \frac{49}{4}\).
5. Rewrite the left side as a perfect square trinomial: \(\left(x - \frac{7}{2}\right)^2 = -12 + \frac{49}{4}\). Then simplify the right side and proceed to solve for \(x\) by taking the square root of both sides.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves adding and subtracting a specific value to both sides to create a binomial squared, making it easier to solve for the variable.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. Understanding its structure is essential for applying methods like completing the square, factoring, or using the quadratic formula to find the roots.

Solving Equations by Isolating the Variable

After forming a perfect square, the next step is to isolate the variable by taking the square root of both sides. This process includes considering both positive and negative roots, which leads to the complete set of solutions for the quadratic equation.