Problem 35 - Algebra 1

Question

Decide how many solutions the equation has. $$-3 x^{2}-5 x+1=0$$

Step-by-Step Solution

Answer

The equation has two real solutions.

Step 1: Identify a, b and c

In the given equation \(-3x^2 -5x + 1 = 0\), a is -3, b is -5, and c is 1.

Step 2: Calculate the Discriminant

The discriminant is calculated by the formula \( D = b^2 - 4ac \). Substituting the identified values to the formula, We get \( (-5)^2 - 4*(-3)*1 = 25 + 12 = 37 \)

Step 3: Determine the Number of Solutions

The value of the discriminant is 37, which is more than zero, therefore, the given equation has two real solutions.

Key Concepts

Discriminant

In quadratic equations, the discriminant is a crucial component for determining the nature of the solutions. It is part of the quadratic formula, denoted as \( D \) in the expression \( D = b^2 - 4ac \). Here, \( a \), \( b \), and \( c \) are coefficients of the terms from the quadratic equation \( ax^2 + bx + c = 0 \).

The discriminant helps us classify the roots, letting us know if they are real or complex, and if there are one or two of them.

Let's explore its significance:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one real solution, sometimes called a repeated or double root.
• If \( D < 0 \), the solutions are complex or imaginary; this means they are not real numbers.

When working with quadratic equations, always calculate the discriminant to understand the nature of the roots efficiently.

Real Solutions

Real solutions in a quadratic equation refer to the values of \( x \) that satisfy the equation and are real numbers.

Whether the solutions are real is determined by the discriminant, \( D = b^2 - 4ac \). If \( D > 0 \), it signifies the equation has two real solutions. These are the points where the parabola represented by the equation crosses the \( x \)-axis.

In our specific problem, the discriminant is calculated to be 37, which is greater than zero. Hence, this confirms that there are indeed two real solutions for the equation. Knowing this is important for guiding you in solving for the value of \( x \), usually by applying the quadratic formula:

• The quadratic formula is \( x = \frac{{-b \pm \sqrt{D}}}{{2a}} \).

By computing \( \sqrt{D} \) and substituting back into the formula, we find those real solutions.

Number of Solutions

The number of solutions of a quadratic equation is fundamentally tied to the value of the discriminant. The clue lies in its calculation: \( D = b^2 - 4ac \).

As discussed, based on \( D \), we can determine:

• Two solutions when \( D > 0 \).
• One solution when \( D = 0 \).
• No real solutions when \( D < 0 \).

In the given equation \(-3x^2 -5x + 1 = 0\), we calculated \( D \) to be 37, thus predicting two solutions. This not only answers how many solutions the equation has but also gives insight into what type of solutions these are, confirming the solutions are real and distinct.