Solve each equation using completing the square. See Examples 3 and 4. 2x^2 + x = 10

Verified step by step guidance

1. Start by moving the constant term to the other side of the equation: \(2x^2 + x - 10 = 0\).
2. Divide every term by the coefficient of \(x^2\), which is 2, to simplify: \(x^2 + \frac{1}{2}x = 5\).
3. To complete the square, take half of the coefficient of \(x\), which is \(\frac{1}{2}\), square it, and add it to both sides: \(x^2 + \frac{1}{2}x + \left(\frac{1}{4}\right)^2 = 5 + \left(\frac{1}{4}\right)^2\).
4. Rewrite the left side as a perfect square trinomial: \((x + \frac{1}{4})^2 = 5 + \frac{1}{16}\).
5. Solve for \(x\) by taking the square root of both sides and then isolating \(x\): \(x + \frac{1}{4} = \pm \sqrt{5 + \frac{1}{16}}\).

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 rearranging the equation and adding a specific value to both sides to create a square of a binomial. This technique simplifies the process of finding the roots of the equation and is particularly useful when the quadratic is not easily factorable.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not zero. The solutions to these equations can be found using various methods, including factoring, using the quadratic formula, or completing the square. Understanding the standard form of a quadratic equation is essential for applying these methods effectively.

Standard Form of a Quadratic

The standard form of a quadratic equation is expressed as y = ax^2 + bx + c. This form allows for easy identification of the coefficients a, b, and c, which are crucial for determining the properties of the parabola represented by the equation, such as its vertex, axis of symmetry, and direction of opening. Recognizing this form is vital for applying techniques like completing the square.