Problem 23 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible. $$2 x^{2}-x=1$$

Step-by-Step Solution

Answer

The solutions to the equation \(2x^2-x=1\) are \(x = 1\) and \(x = -0.5\).

Step 1: Rearrange the Equation

The given equation can be rearranged into the standard form of a quadratic equation \(ax^2+bx+c=0\). Here, a=2, b=-1 and c=-1 so, the equation becomes \(2x^2-x-1=0\).

Step 2: Determination of the Roots

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots of the quadratic equation. Substitute a=2, b=-1, c=-1 into the formula to get \(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4*2*-1}}{2*2}\), which further simplifies to \(x = \frac{1 \pm \sqrt{1+8}}{4}\). After calculating the square root, the equation becomes \(x = \frac{1 \pm \sqrt{9}}{4}\).

Step 3: Simplify the roots

Now simplify the roots by solving each part of the equation. So, the roots of \(x\) will be \(x = \frac{1+3}{4}\) or \(x = \frac{1-3}{4}\). Thus, the roots will be \(x = 1\) or \(x = -0.5\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used for solving quadratic equations, which have the general form \(ax^2 + bx + c = 0\). The formula itself is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps us find the roots of the equation, which are the values of \(x\) that make the equation true.

• \(b^2 - 4ac\) is called the discriminant. It tells us about the nature of the roots:
• If \(b^2 - 4ac > 0\), there are two real and distinct roots.
• If \(b^2 - 4ac = 0\), there is one real double root.
• If \(b^2 - 4ac < 0\), the roots are complex.

By substituting the values of \(a\), \(b\), and \(c\) from our equation into the quadratic formula, we can find the roots easily. It’s a universal method that works for any quadratic equation.

Simplifying Equations

Simplifying equations involves reducing an expression or equation to its simplest form. This makes it easier to solve or understand. In the context of quadratic equations, simplification often involves:

• Combining like terms: Making sure terms with the same variable are added or subtracted correctly.
• Factoring: If applicable, rewriting the quadratic equation as a product of its linear factors.
• Using the quadratic formula: This often includes simplifying the expression within and outside the square root.

For example, in our equation \(2x^2 - x - 1 = 0\), once rearranged, it's fully simplified. Then, during the application of the quadratic formula, solving the inner square root \(\sqrt{b^2 - 4ac}\) is a key point of simplification, helping to further understand the equation.

Roots of Equations

The roots of an equation, also known as solutions, are the values of the variable that satisfy the equation. For quadratic equations, these roots may be real or complex, as indicated by the discriminant \(b^2 - 4ac\).

• Real Roots: These occur when the discriminant is zero or positive, allowing for square root calculations.
• Complex Roots: These arise when the discriminant is negative, as the square root of a negative number involves imaginary units.

In our exercise, the discriminant was positive, and after solving with the quadratic formula, the roots calculated were \( x = 1 \) and \( x = -0.5 \). These are the values where the original equation holds true if you substitute them back into \(2x^2 - x - 1 = 0\). Understanding roots is crucial for visualizing the points where a quadratic curve intersects the x-axis in a graph.