Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to determine the number of real solutions of the quadratic equation. $$2 x^{2}-x-1=0$$

Step-by-Step Solution

Answer

The given quadratic equation has two real solutions.

Step 1: Identify the coefficients of the equation

In the given quadratic equation \(2x^{2} -x - 1=0\), the coefficients are \(a=2\), \(b=-1\) and \(c=-1\).

Step 2: Calculate the discriminant

The discriminant \(\Delta\) is calculated using the formula \(\Delta = b^{2} - 4ac\). Therefore, \(\Delta = (-1)^{2} - 4*2*(-1) = 1 + 8 = 9\).

Step 3: Determine the number of real solutions

Since the value of the discriminant \(\Delta\) is greater than 0 (\(\Delta = 9\)), the given quadratic equation has two real solutions.

Key Concepts

Using a Graphing Utility

When looking at a quadratic equation like \(2x^{2}-x-1=0\), a graphing utility can be an excellent tool for visual learners who need to understand its real solutions. A graphing utility can plot the equation which is represented typically by a parabolic curve. The points where this curve crosses the x-axis are the real solutions to the equation.

For \(2x^{2}-x-1=0\), after plotting, you would see the parabola crossing the x-axis twice, suggesting that there are two real solutions. This visual method confirms the analytical approach that shows the same result. It's highly beneficial to compare both graphing and calculation to reinforce the understanding of how quadratic equations function.

Understanding the Discriminant

The discriminant is a powerful tool in determining the nature of the solutions of a quadratic equation, without necessarily solving the equation itself. It is represented by the symbol \(\Delta\) and is calculated from the coefficients of the quadratic equation in the form \(ax^{2} + bx + c = 0\).

The discriminant formula is \(\Delta = b^{2} - 4ac\). Depending on the value of \(\Delta\), we can predict:

• \(\Delta > 0\): Two distinct real solutions (the parabola crosses the x-axis at two points).
• \(\Delta = 0\): One real solution (the parabola touches the x-axis at one point).
• \(\Delta < 0\): No real solutions (the parabola does not cross the x-axis).

For our given equation \(2x^{2} - x - 1 = 0\), computing \(\Delta = (-1)^{2} - 4\cdot 2 \cdot (-1)\) gives us 9, which is greater than 0. This means there are two real solutions.

Applying the Quadratic Formula

The quadratic formula is a well-established method for solving equations of the form \(ax^{2} + bx + c = 0\). It states that the solutions for \(x\) are given by the expression \(\frac{-b \pm \sqrt{\Delta}}{2a}\), where \(\Delta\) is the discriminant.

When applying this formula to our equation \(2x^{2}-x-1=0\), we use the already calculated discriminant, 9, and plug in the values for \(a\), \(b\), and \(c\) from the coefficients. Through this process, we will arrive at the two real solutions of our quadratic equation. It’s important to be comfortable calculating the discriminant, as it’s a crucial step in the formula that tells us not only the quantity but also allows us to compute the actual solutions.