Problem 15 - 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. \(-12 x^{2}+5 x+2=0\)

Step-by-Step Solution

Answer

Discriminant is 121; two distinct real roots: \(-\frac{1}{4}\) and \(\frac{2}{3}\).

Step 1: Identify the coefficients

The given quadratic equation is \(-12x^2 + 5x + 2 = 0\). Here, the coefficients are: - \(a = -12\) - \(b = 5\) - \(c = 2\).

Step 2: Calculate the discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). Substituting the values, we have:\[D = 5^2 - 4(-12)(2)\]\[D = 25 + 96 = 121\].

Step 3: Interpret the discriminant

The discriminant \(D = 121\) is positive. This indicates that the quadratic equation has two distinct real roots.

Step 4: Apply the Quadratic Formula

The Quadratic Formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). With \(a = -12\), \(b = 5\), and \(D = 121\):\[x = \frac{-5 \pm \sqrt{121}}{2(-12)}\]\[x = \frac{-5 \pm 11}{-24}\].

Step 5: Calculate the roots

First, calculate \(x = \frac{-5 + 11}{-24}\):\[x = \frac{6}{-24} = -\frac{1}{4}\].Next, calculate \(x = \frac{-5 - 11}{-24}\):\[x = \frac{-16}{-24} = \frac{2}{3}\].

Key Concepts

Discriminant

The discriminant is a key part of quadratic equations. It's found inside the quadratic formula and tells us about the nature of the roots of the equation. For any quadratic equation given by the general form \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula:

• \(D = b^2 - 4ac\)

The discriminant helps us know the type and number of solutions the equation has:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root (also called a repeated or double root).
• If \(D < 0\), there are no real roots; instead, there are two complex roots.

In our example \(-12x^2 + 5x + 2 = 0\), the discriminant is 121, which means this equation has two distinct real roots.

Quadratic Formula

The Quadratic Formula is a powerful tool for solving any quadratic equation, regardless of its complexity. The formula is:

• \(x = \frac{-b \pm \sqrt{D}}{2a}\)

In this formula, \(b\) and \(a\) are coefficients from the equation, and \(\sqrt{D}\) is the square root of the discriminant. The plus-minus symbol (\(\pm\)) suggests that we solve the equation twice: once using the plus sign and once using the minus sign.
This dual calculation explains why we find up to two roots. Applying this formula allows us to find exact solutions for the roots of the equation. It is especially helpful when factoring isn't convenient or possible.

Real Roots

Real roots are the solutions to a quadratic equation that you can plot on the number line. They are not imaginary or complex, which means they do not include the imaginary unit \(i\). Real roots are the x-values where the graph of the quadratic equation crosses the x-axis.
For the quadratic equation \(-12x^2 + 5x + 2 = 0\), since the discriminant is positive (\(D = 121\)), we find there are two distinct real roots.
Through the quadratic formula, these roots are calculated to be:

• \(-\frac{1}{4}\)
• \(\frac{2}{3}\)

Finding real roots is crucial in many applications, including physics and engineering, where real solutions often represent measurable quantities.

Coefficients of Quadratic Equation

A quadratic equation is an expression where the highest degree of a variable is 2. It typically takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients.

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

Each coefficient plays a significant role:

• \(a\) dictates the "width" and direction of the parabola (narrower if \(|a|\) is large, wider if small). If \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards.
• \(b\) and \(c\) influence the position of the parabola along the x and y axes.

For \(-12x^2 + 5x + 2 = 0\), we identified \(a = -12\), \(b = 5\), and \(c = 2\). These values directly impact the discriminant and the resulting roots when applying the quadratic formula.