Problem 9 - Calculus: One and Several Variables

Question

Find the real roots of the equation. $$2 x^{2}+x-1=0$$

Step-by-Step Solution

Answer

The real roots of the equation \(2x^2 + x - 1 = 0\) are \(x = \frac{1}{2}\) and \(x = -1\).

Step 1: Identify Coefficients

The given quadratic equation is \(2x^2 + x - 1 = 0\). The coefficients are therefore \(a = 2\), \(b = 1\) and \(c = -1\).

Step 2: Substitute Coefficients in the Quadratic Formula

Substitute \(a\), \(b\), and \(c\) into the quadratic formula. This means we have \(x = \frac{-1 \pm \sqrt{1^2 - 4*2*(-1)}}{2*2}\).

Step 3: Simplify the Equation

Simplify the equation to find two possible solutions for \(x\). After simplifying, you will get \(x = \frac{-1 \pm \sqrt{9}}{4}\). Then further simplify to get \(x = \frac{-1 \pm 3}{4}\).

Step 4: Determine the Real Roots

The real roots of the equation are the results of the simplified equation when \(\pm\) is replaced with \(+\) and \(-\).The solutions therefore will be \(x = \frac{1}{2}\) and \(x = -1\).

Key Concepts

Real Roots

When working with quadratic equations, finding the real roots is a fundamental task. Real roots are simply the x-values where the graph of the quadratic equation intersects the x-axis. In mathematical terms, these are solutions to the equation and can be obtained using various methods.
For a quadratic equation in standard form, usually given as \(ax^2 + bx + c = 0\), the roots can be determined by calculating where the equation equals zero. Depending on the value under the square root in the quadratic formula, you can have:

• Two real roots, if the value (discriminant) is positive.
• One real root, if the value is zero.
• No real roots if the value is negative, resulting in complex or imaginary roots.

In our exercise, the quadratic equation had real roots of \(x = \frac{1}{2}\) and \(x = -1\), since our calculations resulted in positive values under the square root.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. This formula is especially useful when factoring is difficult or impossible. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula will give you the exact solutions for any quadratic equation \(ax^2 + bx + c = 0\), assuming the roots are real numbers.
It involves:

• Calculating the discriminant, \(b^2 - 4ac\), which reveals the nature of the roots.
• The operations of addition or subtraction after finding the square root of the discriminant.
• Finally, division by \(2a\) to determine each potential root.

For our problem \(2x^2 + x - 1 = 0\), substituting the coefficients into the formula provided us with the roots \(x = \frac{1}{2}\) and \(x = -1\). Using the quadratic formula ensures we accurately identify these real roots.

Simplifying Equations

After substituting the values into the quadratic formula, the next step is simplifying the equation. This involves dealing with any square roots and arithmetic operations to bring the equation to its simplest form.
To simplify, you must:

• First calculate the discriminant \(b^2 - 4ac\) and find its square root.
• Then, solve for \(x\) by performing the addition and subtraction operations specified by the formula.
• Simplify fractions or any fraction multiplication that occurs in the result.

In the original problem, the simplification process led us from \(x = \frac{-1 \pm \sqrt{9}}{4}\) to the simpler solutions \(x = \frac{1}{2}\) and \(x = -1\). Simplification ensures the answer is in its most understandable and usable form.