Problem 48 - Algebra and Trigonometry

Question

Find all real solutions of the equation. \(5 x^{2}-7 x+5=0\)

Step-by-Step Solution

Answer

No real solutions.

Step 1: Identify a, b, and c

The equation is quadratic and in standard form, \(ax^2 + bx + c = 0\). From the given equation \(5x^2 - 7x + 5 = 0\), identify the coefficients: \(a = 5\), \(b = -7\), and \(c = 5\).

Step 2: Calculate the Discriminant

The discriminant \( \Delta \) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \( \Delta = b^2 - 4ac \). Substitute the values of \(a\), \(b\), and \(c\):\[ \Delta = (-7)^2 - 4\cdot5\cdot5 = 49 - 100 = -51 \] Since the discriminant is negative, the equation has no real solutions.

Key Concepts

Real Solutions

A quadratic equation like the one in our exercise can have different types of solutions. Of particular interest are 'real solutions,' which are the solutions that do not involve imaginary numbers. Real solutions are numbers that can be plotted on the number line and are therefore "real" in that sense.

Quadratic equations can have:

• Two distinct real solutions
• One real solution (a repeated or double root)
• No real solutions

In the context of the exercise given, whether or not the equation has real solutions depends primarily on the value of the discriminant. Understanding the discriminant is key to determining the nature of these solutions.

Discriminant

The discriminant of a quadratic equation plays a crucial role in determining the nature and number of solutions of the equation. For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is calculated using the formula:

• \(\Delta = b^2 - 4ac\)

The discriminant value tells us the nature of the roots:

• If \(\Delta > 0\): Two distinct real solutions exist.
• If \(\Delta = 0\): Exactly one real solution exists (the roots are repeated).
• If \(\Delta < 0\): No real solutions exist; the roots are complex or imaginary.

In the exercise, we found that \(\Delta = -51\), which is less than zero, indicating that there are no real solutions. This helps us identify our next step, which is potentially exploring complex solutions if needed.

Quadratic Formula

The quadratic formula is a universal tool used to find the solutions of any quadratic equation. For an equation \(ax^2+bx+c=0\), the solutions can be found using:

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

This equation relies heavily on the discriminant \(\Delta = b^2 - 4ac\), as it appears under the square root. Therefore, the discriminant directly influences the type of solutions:

• Two real solutions if \(\Delta > 0\).
• One real solution if \(\Delta = 0\).
• Two complex solutions if \(\Delta < 0\).

In our scenario, given that \(\Delta = -51\), the square root of a negative number gives us complex solutions. The quadratic formula confirms there are no real solutions for this equation, which is consistent with our findings from the discriminant analysis. This formula is essential in solving quadratics and is typically the go-to method when manual factoring isn't feasible.