Problem 82 - Algebra 1: Concepts and Skills

Question

Tell whether the equation has two solutions, one solution, or no real solution. $$ 3 x^{2}-7 x+5=0 $$

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Identify a, b and c

From the given equation, \(3x^2 - 7x +5 = 0\), we identify the coefficients; \(a = 3\), \(b = -7\) and \(c = 5\).

Step 2: Compute the discriminant

Use the formula for the discriminant which is \(b^2 - 4ac\). Substituting the identified values, we get \((-7)^2 - 4*3*5 = 49 - 60 = -11\).

Step 3: Determine the number of solutions

Since the discriminant is negative (-11), it means that the equation has no real solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are mathematical expressions of the form \( ax^2 + bx + c = 0 \). This specific formula is derived from completing the square in the general quadratic equation and is given by:

\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]

In this formula, \(a\), \(b\), and \(c\) represent the coefficients of the quadratic equation, where \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term. The symbol \(\pm\) indicates that there are generally two solutions, one for \(+\) and one for \(–\), corresponding to the roots of the equation. The term under the square root, \(b^2 - 4ac\), is known as the discriminant and it plays a crucial role in determining the nature and number of roots which leads us to our next concept.

Nature of Roots

The discriminant in a quadratic equation not only facilitates in finding the roots but also reveals their nature. To simply put, the value of the discriminant tells us whether the roots are real or complex, and if they're real, whether they are distinct or repeated.

The discriminant is calculated using the formula \(b^2 - 4ac\). Depending on its value, we have:

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the equation has exactly one real root, also known as a repeated or double root.
• If \(b^2 - 4ac < 0\), there are no real solutions; rather, the equation has two complex roots.

For the exercise in question, with \(a = 3\), \(b = -7\), and \(c = 5\), the discriminant is \(-11\), which is less than zero. Thus, the equation has no real solutions and instead has two complex roots.

Solving Quadratic Equations

Solving quadratic equations can be done using a variety of methods such as factoring, graphing, completing the square, and using the quadratic formula, which is often the most direct approach. To solve a quadratic equation, one must first bring the equation to the standard form \(ax^2 + bx + c = 0\), identifying the coefficients \(a\), \(b\), and \(c\). Then, based on the discriminant \(b^2 - 4ac\), the solver can decide the best course of action.

If the discriminant is positive, the equation can be solved using the quadratic formula directly to find two distinct real roots. When the discriminant is zero, the formula will yield one real solution. In cases where the discriminant is negative, it indicates that the equation has complex roots and one must include the imaginary unit \(i\) to represent the square root of \(–1\) in the solutions.

Solving the given exercise using the quadratic formula is not necessary because the negative discriminant immediately suggests that the roots are complex and cannot be simplified further using real numbers.