Problem 16 - Algebra 2

Question

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(-3 x^{2}-5 x+2=0\)

Step-by-Step Solution

Answer

Discriminant is 49; two distinct real roots; solutions are \(-2\) and \(\frac{1}{3}\).

Step 1: Identify coefficients

For the quadratic equation given, \(-3x^2 - 5x + 2 = 0\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = -3\), \(b = -5\), and \(c = 2\).

Step 2: Find the Discriminant

The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Substitute the coefficients: \(D = (-5)^2 - 4(-3)(2) = 25 + 24 = 49\).

Step 3: Determine the Type of Roots

Based on the discriminant \(D = 49\), determine the number and type of roots. Since \(D > 0\), the equation has two distinct real roots.

Step 4: Use the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to find the exact solutions. Substitute \(b = -5\), \(a = -3\), and \(D = 49\): \(x = \frac{-(-5) \pm \sqrt{49}}{2(-3)}\).

Step 5: Simplify the Solution

Simplify the expression for \(x\): \(x = \frac{5 \pm 7}{-6}\). This calculation yields two roots: \(x = \frac{5 + 7}{-6} = -2\) and \(x = \frac{5 - 7}{-6} = \frac{1}{3}\).

Key Concepts

Discriminant

The discriminant is a crucial concept in quadratic equations. It helps determine the nature of the roots of the equation. For any quadratic equation in the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is given by the formula:

• \(D = b^2 - 4ac\)

In this case, for the equation \(-3x^2 - 5x + 2 = 0\), we substitute the coefficients \(a = -3\), \(b = -5\), and \(c = 2\), into the discriminant formula.
This results in:

• \(D = (-5)^2 - 4(-3)(2) = 25 + 24 = 49\)

The value of the discriminant provides insight about the roots without solving the equation. It tells us if the roots are real or complex, and whether the roots are distinct or repeated.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:

• \(x = \frac{-b \pm \sqrt{D}}{2a}\)

Here, \(D\) is the discriminant, \(b\) is the coefficient of the \(x\) term, and \(a\) is the coefficient of the \(x^2\) term. The plus-minus sign \(\pm\) indicates that there will generally be two solutions or roots.
Let's apply the quadratic formula to our specific equation, \(-3x^2 - 5x + 2 = 0\). We already found that \(D = 49\). Substituting \(a = -3\), \(b = -5\), and \(D = 49\) into the formula, we calculate:

• \(x = \frac{-(-5) \pm \sqrt{49}}{2(-3)}\)
• \(x = \frac{5 \pm 7}{-6}\)

With this, we can proceed to simplify and find the exact solutions, highlighting the ease of use and general applicability of the quadratic formula.

Real Roots

Understanding the nature of the roots of a quadratic equation helps predict not only the number of solutions but also their types. When the discriminant \(D\) is positive, as in our case with \(D = 49\), it signifies two distinct real roots.
These roots are the values of \(x\) that satisfy the equation, causing it to equal zero. Real roots indicate that where a graph of the equation would intersect or "cross" the x-axis.
For our equation, the solutions we obtain by substituting the discriminant in the quadratic formula are real numbers:

• \(x_1 = \frac{5 + 7}{-6} = -2\)
• \(x_2 = \frac{5 - 7}{-6} = \frac{1}{3}\)

Both roots \(-2\) and \(\frac{1}{3}\) can be clearly plotted on the number line, demonstrating they exist within the real number system and providing tangible interpretations for the resulting quadratic.

Coefficient Identification

Before solving a quadratic equation using either the quadratic formula or any other method, identifying the coefficients is crucial. The typical quadratic equation is structured as \(ax^2 + bx + c = 0\). Here:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

For our given equation \(-3x^2 - 5x + 2 = 0\), the coefficients are \(a = -3\), \(b = -5\), and \(c = 2\).
Identifying these coefficients is the first step in the process as it directs the subsequent calculation of the discriminant and application of the quadratic formula.
Clear identification ensures accuracy and helps clarify each element's role in determining the equation's roots, laying the groundwork for further mathematical exploration or graphing.