Problem 19 - 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. \(x^{2}+3 x-3=0\)

Step-by-Step Solution

Answer

The roots are real, distinct, and given by \(x = \frac{-3 \pm \sqrt{21}}{2}\).

Step 1: Identify coefficients

In the quadratic equation \(x^2 + 3x - 3 = 0\), identify the coefficients: \(a = 1\), \(b = 3\), and \(c = -3\).

Step 2: Calculate the Discriminant

The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(D = 3^2 - 4(1)(-3)\). This simplifies to \(D = 9 + 12 = 21\).

Step 3: Analyze the Discriminant

Since the discriminant \(D = 21\) is greater than zero, the quadratic equation has two distinct real roots.

Step 4: Apply the Quadratic Formula

Use the Quadratic Formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to find the roots. Substitute \(a = 1\), \(b = 3\), and \(D = 21\) into the formula: \(x = \frac{-3 \pm \sqrt{21}}{2}\).

Step 5: Simplify to Find Exact Solutions

Simplify the expression \(x = \frac{-3 + \sqrt{21}}{2}\) and \(x = \frac{-3 - \sqrt{21}}{2}\) to get the exact solutions.

Key Concepts

Discriminant

The discriminant is a key component in solving quadratic equations. It's derived from the quadratic formula, but its specific role is to help us determine the nature of the roots of the equation. The discriminant is given by the expression:
\[ D = b^2 - 4ac \]where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). This value is crucial because:

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root, meaning the roots are repeated or a double root.
• If \( D < 0 \), the equation has no real roots, only complex ones.

In the example given, the quadratic equation \( x^2 + 3x - 3 = 0 \) was evaluated to have a discriminant \( D = 21 \), indicating two distinct real roots.

Quadratic Formula

The quadratic formula is a universal method to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). This formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]It provides an exact solution to quadratic equations by using the coefficients \( a \), \( b \), and \( c \). Here's how the formula works:

• Compute the discriminant, \( D = b^2 - 4ac \), which is already part of the formula under the square root.
• The formula uses \( \pm \) to incorporate both potential solutions for \( x \), giving two roots when the discriminant is positive.
• This is a reliable method that bypasses the need for factoring or completing the square, especially useful when the roots are not easy integers.

In the discussed example, substituting \( a = 1 \), \( b = 3 \), and \( D = 21 \) into the formula yields the two roots: \( x = \frac{-3 \pm \sqrt{21}}{2} \).

Roots of a Quadratic Equation

Finding the roots of a quadratic equation is the main goal when dealing with these expressions. The roots, also known as solutions, are the values of \( x \) for which the equation \( ax^2 + bx + c = 0 \) equals zero. These values can tell us a lot about the graph of the quadratic:

• If there are two distinct real roots, the parabola will intersect the x-axis at two points.
• One repeated real root means the parabola just touches the x-axis, indicating a vertex on the axis (a perfect square).
• Complex roots will result in a parabola that does not intersect the x-axis, remaining entirely above or below it.

In the example equation \( x^2 + 3x - 3 = 0 \), the roots found using the quadratic formula are \( x = \frac{-3 + \sqrt{21}}{2} \) and \( x = \frac{-3 - \sqrt{21}}{2} \). These roots signify where the parabola cuts across the x-axis, offering a graphical insight into the solutions.