Solve each equation. 2x²+x-15 = 0

Verified step by step guidance

1. Identify the quadratic equation in the standard form: \( ax^2 + bx + c = 0 \). Here, \( a = 2 \), \( b = 1 \), and \( c = -15 \).
2. Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
3. Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula: \( x = \frac{-(1) \pm \sqrt{(1)^2 - 4(2)(-15)}}{2(2)} \).
4. Simplify the expression under the square root (the discriminant): \( 1^2 - 4 \times 2 \times (-15) \).
5. Calculate the two possible values for \( x \) using the plus and minus in the quadratic formula: \( x = \frac{-1 + \sqrt{\text{discriminant}}}{4} \) and \( x = \frac{-1 - \sqrt{\text{discriminant}}}{4} \).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. For quadratic equations, this often involves finding two binomials that multiply to give the quadratic. Mastery of factoring techniques is crucial for quickly solving equations like 2x² + x - 15 = 0.

The Quadratic Formula

The quadratic formula, given by x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of any quadratic equation. This formula is particularly useful when the equation cannot be easily factored. Understanding how to apply the quadratic formula allows students to solve for x in a systematic way, regardless of the complexity of the coefficients.