Problem 18 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$-9 x^{2}+24 x-16=0$$

Step-by-Step Solution

Answer

\(x = \frac{4}{3}\)

Step 1: Identify the coefficients

The quadratic equation is given in the form \begin{center}$$ax^2 + bx + c = 0$$\temphasizing an informal approach. Identify the coefficients in the equation \begin{center}$$-9x^2 + 24x - 16 = 0$$. Here, $$a = -9$$, $$b = 24$$, and $$c = -16$$.

Step 2: Write down the quadratic formula

The quadratic formula is \begin{center}$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.

Step 3: Calculate the discriminant

The discriminant \begin{center}$$b^2 - 4ac$$. Substitute in the values: $$b = 24$$, $$a = -9$$, and $$c = -16$$. Calculate: $$24^2 - 4(-9)(-16) = 576 - 576 = 0$$. Thus, the discriminant is zero.

Step 4: Calculate the roots

With a zero discriminant, there is one unique solution.\begin{center}Substitute back into the quadratic formula: $$x = \frac{-24 \pm 0}{2(-9)} = \frac{-24}{-18} = \frac{4}{3}$$.

Step 5: State the solution

Therefore, the solution to the quadratic equation \begin{center}$$-9x^2 + 24x - 16 = 0$$ is: \begin{center}$$x = \frac{4}{3}$$.

Key Concepts

Discriminant

The discriminant in a quadratic equation plays an important role in determining the nature of the roots. It is the part under the square root in the quadratic formula, given by the expression \(b^2 - 4ac\). To find the discriminant, you need to identify the values of the coefficients \(a\), \(b\), and \(c\) from the general form \(ax^2 + bx + c = 0\). Plug these values into the formula and calculate. For example, in the equation \(-9x^2 + 24x - 16 = 0\), the discriminant is calculated as follows:
\24^2 - 4(-9)(-16) = 576 - 576 = 0\.
If the discriminant is positive, there are two distinct real roots.
If it is zero, as in our example, there is exactly one real root.
If the discriminant is negative, there are no real roots, but two complex roots instead.

Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable \(x\). It is typically of the form \(ax^2 + bx + c = 0\). This means it has three parts:

• \(ax^2\) is the quadratic term.
• \(bx\) is the linear term.
• \(c\) is the constant term.

The coefficients \(a\), \(b\), and \(c\) can be any real numbers, but \(a\) should never be zero, otherwise the equation is not quadratic.
To solve a quadratic equation, you can use various methods like factoring, completing the square, or the quadratic formula. The quadratic formula, \x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\, is the most universal method because it works for all quadratic equations regardless of the type of roots.

Roots of an Equation

The roots of a quadratic equation are the values of \(x\) that make the equation equal to zero. These are also known as the solutions or the zeroes of the equation. By applying the quadratic formula, you can find these roots easily.
The quadratic formula provides two roots, given as follows: \x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\ and \x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\.
The discriminant \(b^2 - 4ac\) decides the nature of the roots:

• When the discriminant is positive, you get two distinct real roots.
• When the discriminant is zero, you get one unique real root.
• When the discriminant is negative, you get two complex roots.

In our example, the discriminant was zero, so there was exactly one solution for the equation \( -9x^2 + 24x - 16 = 0 \), which is \x = \frac{4}{3}\.