Problem 56 - Algebra and Trigonometry

Question

Solve: \(\left.3 x^{2}+1=x^{2}+x . \text { (Section } 1.5, \text { Example } 8\right)\)

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = \frac{1 + i\sqrt{7}}{4}\) and \(x = \frac{1 - i\sqrt{7}}{4}\).

Step 1: Combine Like Terms

Subtract \(x^2\) and \(x\) from both sides to set the equation to zero, which gives the result: \(2x^2 - x + 1 = 0\).

Step 2: Apply the Quadratic Formula

The quadratic formula is \(x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\). Apply this formula using a=2, b=-1 and c=1. This gives: \(x = \frac{-(-1) ± \sqrt{(-1)^2 - 4*2*1}}{2*2}\) which simplifies to \(x = \frac{1 ± \sqrt{1 - 8}}{4}\).

Step 3: Simplify the expression

Simplify the expression in the radical, giving: \(x = \frac{1 ± \sqrt{-7}}{4}\)

Step 4: Presenting the Solution

Since the solution involves the square root of a negative number, it involves imaginary or complex numbers. We write \(\sqrt{-1}\) as \(i\). As such, the solution becomes \(x = \frac{1 ± i\sqrt{7}}{4}\).

Key Concepts

Quadratic Formula

Understanding the Quadratic Formula is crucial for solving quadratic equations. It provides a straightforward method for finding the roots of any quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\). The formula itself is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the equation.

Applying this formula to the given problem, with \(a = 2\), \(b = -1\), and \(c = 1\), allows us to find the solutions for the variable \(x\). The step-by-step process shows us substituting these values into the formula, leading to \(x = \frac{1 \pm \sqrt{1 - 8}}{4}\), with the expression under the square root (\