Problem 62 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation using any method. Find only real solutions. $$-x^{2}-3 x=1$$

Step-by-Step Solution

Answer

The real solutions to the equation are \(x_{1} = -0.382\) and \(x_{2} = -2.618\)

Step 1: Rearranging the equation.

The first step is to rearrange the equation in the form \(ax^{2}+bx+c=0\). Therefore, you can rearrange the equation as follows: \(x^{2}+3x+1=0\)

Step 2: Applying the Quadratic Formula

Apply the quadratic formula which is \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Here, a=1, b=3 and c=1.

Step 3: Calculating the discriminant

Calculate the discriminant (\(b^{2}-4ac\)). Let's substitute a, b and c to find the discriminant: \(3^{2}-4(1)(1) = 9 -4 = 5 \)

Step 4: Calculating the roots

Next, calculate the roots by substituting a, b, and the discriminant into the quadratic formula from step 2: \( x=\frac{-3 \pm \sqrt{5}}{2} \) which simplifies to \(x= -1.5 \pm 1.118\)

Step 5: Finding the solutions

Lastly, calculate the values for x: \( x_{1} = -1.5 + 1.118 = -0.382, x_{2} = -1.5 - 1.118 = -2.618 \)

Key Concepts

Understanding the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations in the form \(ax^2 + bx + c = 0\). This formula allows us to find the values of \(x\) that make the equation true. The formula itself is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Each part of the formula plays a crucial role:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
• The term \(-b\) ensures that the solution takes into account the sign of \(b\), leading to two potential solutions due to the \(\pm\) sign.
• The square root part, \(\sqrt{b^2 - 4ac}\), influences whether the solutions are real numbers or not, which we'll explore in the discriminant section.
• The division by \(2a\) normalizes the equation based on the leading coefficient \(a\).

By applying this formula, you can systematically find solutions to any quadratic equation, provided you handle each component correctly.

Role and Importance of the Discriminant

The discriminant is a special part of the quadratic formula, represented by \(b^2 - 4ac\). It determines the nature of the solutions of the quadratic equation. Understanding the discriminant is key to predicting how many solutions a quadratic equation has and whether they are real or complex.

• If the discriminant is greater than 0, there are two distinct real solutions. This occurs when the graph of the quadratic equation intersects the x-axis at two points.
• When the discriminant equals 0, there is exactly one real solution, also known as a repeated root. This means the graph touches the x-axis at a single point.
• If the discriminant is less than 0, there are no real solutions, and instead, we have two complex solutions. This means the graph does not intersect the x-axis.

In the original exercise, the discriminant was calculated as 5, which is greater than 0. Consequently, the equation has two distinct real solutions.

Finding and Recognizing Real Solutions

Real solutions to a quadratic equation are values of \(x\) where the graph meets the x-axis, signifying actual intersections that you can plot. Identifying these involves both solving mathematically and interpreting the discriminant.Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), find:

• \(x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\) and
• \(x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\)

In practical scenarios, after determining the discriminant is positive, use the quadratic formula to solve for \(x\). Calculations will show each solution distinctly, just as seen in the exercise:

• \(x_1 = -1.5 + 1.118 = -0.382\)
• \(x_2 = -1.5 - 1.118 = -2.618\)

These values are the real solutions where the parabola of the quadratic equation crosses the x-axis bit. Understanding each step and the relevance of these solutions helps build intuitive and practical mathematical skills.