Solve each equation using the quadratic formula. (1/2)x² + (1/4)x - 3 = 0

Verified step by step guidance

1. Identify the coefficients in the quadratic equation \(\frac{1}{2}x^2 + \frac{1}{4}x - 3 = 0\). Here, \(a = \frac{1}{2}\), \(b = \frac{1}{4}\), and \(c = -3\).
2. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula is used to find the roots of any quadratic equation \(ax^2 + bx + c = 0\).
3. Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-\frac{1}{4} \pm \sqrt{\left(\frac{1}{4}\right)^2 - 4 \cdot \frac{1}{2} \cdot (-3)}}{2 \cdot \frac{1}{2}}\).
4. Simplify inside the square root (the discriminant): calculate \(\left(\frac{1}{4}\right)^2\) and \(4 \cdot \frac{1}{2} \cdot (-3)\), then find their sum.
5. Simplify the denominator \(2 \cdot \frac{1}{2}\) and then write the two possible solutions for \(x\) by evaluating \(-\frac{1}{4} \pm \sqrt{\text{discriminant}}\) divided by the denominator.

Key Concepts

Quadratic Equation Standard Form

A quadratic equation is typically written as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Recognizing and rewriting the given equation into this standard form is essential before applying any solution methods.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation. It uses the coefficients a, b, and c from the standard form and calculates roots based on the discriminant.

Discriminant and Nature of Roots

The discriminant, given by b² - 4ac, determines the type of roots of a quadratic equation. If positive, there are two real roots; if zero, one real root; and if negative, two complex roots. This helps interpret the solutions found using the quadratic formula.