Solve each equation. 2x²+x-15 = 0

Verified step by step guidance

1. Identify the quadratic equation in standard form: \$2x^{2} + x - 15 = 0$.
2. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are coefficients from the equation \(ax^{2} + bx + c = 0\).
3. Determine the coefficients: \(a = 2\), \(b = 1\), and \(c = -15\).
4. Calculate the discriminant: \(\Delta = b^{2} - 4ac = 1^{2} - 4 \times 2 \times (-15)\).
5. Substitute the values into the quadratic formula and simplify under the square root and the entire expression to find the two possible values of \(x\).

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and typically has two solutions, which can be real or complex numbers.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. This method is useful when the quadratic can be expressed as (mx + n)(px + q) = 0, allowing the use of the zero-product property to find solutions.

Zero-Product Property

The zero-product property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is essential for solving equations after factoring, as it leads to setting each factor equal to zero to find the roots.