Problem 32 - Contemporary Precalculus

Question

Find the number of real solutions of the equation by computing the discriminant. $$49 t^{2}+5=42 t$$

Step-by-Step Solution

Answer

Answer: The given quadratic equation has two distinct real solutions.

Step 1: Identify Quadratic Coefficients

We are given the quadratic equation: $$49t^2 - 42t + 5 = 0$$ Here, a = 49, b = -42, and c = 5.

Step 2: Compute the Discriminant

Using the formula for the discriminant Δ, we compute the value of Δ as follows: $$Δ = b^2 - 4ac$$ $$Δ = (-42)^2 - 4(49)(5)$$

Step 3: Calculate the Value of the Discriminant

Now, plug in the values of a, b, and c into the formula and calculate the value of Δ: $$Δ = (-42)^2 - 4(49)(5)$$ $$Δ = 1764 - 980$$ $$Δ = 784$$

Step 4: Analyze the Discriminant

Now that we've calculated the value of the discriminant, we can analyze its value to determine the number of real solutions of the given quadratic equation. 1. If Δ > 0, the quadratic equation has two distinct real solutions. 2. If Δ = 0, the quadratic equation has one real solution (a repeated root). 3. If Δ < 0, the quadratic equation has no real solutions (only complex solutions). We've computed the value of Δ as 784, which is greater than 0.

Step 5: Conclusion

As Δ > 0 (Δ = 784), the given quadratic equation has two distinct real solutions.

Key Concepts

Understanding the Discriminant

The discriminant is an important value that helps us determine the number and type of solutions a quadratic equation has. It comes from the quadratic formula and is represented by the symbol \(\Delta\). When you're dealing with a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant formula is used as follows:

• \(\Delta = b^2 - 4ac\)

For the equation \(49t^2 - 42t + 5 = 0\), we identified \(a = 49\), \(b = -42\), and \(c = 5\).By substituting these values into the formula:

• \(\Delta = (-42)^2 - 4(49)(5)\)
• \(\Delta = 1764 - 980\)
• \(\Delta = 784\)

Since \(\Delta = 784\), we can make conclusions about the nature of the solutions.

Exploring Real Solutions

Real solutions of a quadratic equation are the values of the variable that make the equation true. They can be identified by analyzing the value of the discriminant.Here's how we determine the number of real solutions based on the discriminant:

• If \(\Delta > 0\): There are two distinct real solutions. This means that the parabola intersects the x-axis at two points.
• If \(\Delta = 0\): There is one real solution, which is a repeated root. The parabola touches the x-axis at one point, also known as a double root.
• If \(\Delta < 0\): There are no real solutions, meaning the solutions are complex or imaginary. The parabola does not intersect the x-axis.

For the given equation, \(\Delta = 784\) which is greater than zero, so there are two distinct real solutions.

Using the Quadratic Formula

The quadratic formula is a versatile tool used to find the roots of any quadratic equation. It is particularly helpful when factoring is difficult or impossible. The formula is:

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

Once the value of the discriminant \(\Delta = b^2 - 4ac\) is known, you can easily substitute into the quadratic formula to find the real solutions.For the equation \(49t^2 - 42t + 5 = 0\), substituting \(a = 49\), \(b = -42\), and \(c = 5\) gives:

• \(t = \frac{-(-42) \pm \sqrt{784}}{2 \cdot 49}\)
• \(t = \frac{42 \pm 28}{98}\)

This yields two different solutions, confirming that our analysis of the discriminant was correct. Always remember, the discriminant clarifies how many solutions you'll find, while the quadratic formula reveals their exact values.