Problem 57 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. \(p(x)=x^{2}-6 x-16 .\) Find \(x\) so that \(p(x)=0\)

Step-by-Step Solution

Answer

The solutions for the equation \(x^2 - 6x - 16 = 0\) are \(x = {-2, 8}\).

Step 1: Identify the coefficients in the quadratic equation

The given quadratic equation is \(x^2 - 6x - 16 = 0\). The coefficients of this equation are: - a (coefficient of \(x^2\)) = 1 - b (coefficient of x) = -6 - c (constant term) = -16

Step 2: Apply the quadratic formula

To solve the equation, we can use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substitute the values of a, b, and c into the formula: \[x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-16)}}{2(1)}\]

Step 3: Calculate the discriminant

First, we need to calculate the value inside the square root, which is the discriminant: \(\Delta = b^2 - 4ac = (-6)^2 - 4(1)(-16) = 36 + 64 = 100\)

Step 4: Solve for x

Now, we can use the discriminant to solve for x: \[x = \frac{6 \pm \sqrt{100}}{2}\] There are two possible solutions for x, as we have a plus-minus sign in the formula. - For the "+" sign: \(x = \frac{6 + 10}{2} = \frac{16}{2} = 8\) - For the "-" sign: \(x = \frac{6 - 10}{2} = \frac{-4}{2} = -2\) So the solutions for the equation \(x^2 - 6x - 16 = 0\) are \(x = {-2, 8}\).

Key Concepts

Coefficients

When dealing with quadratic equations, it's essential to identify the coefficients properly as they play a crucial role in solving the equation. A quadratic equation is typically of the form \(ax^2 + bx + c = 0\), where:

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

These coefficients determine the shape and position of the parabola represented by the quadratic equation. By substituting these values into various formulas, such as the quadratic formula, we can find the roots of the equation. In the problem provided, the coefficients are \(a = 1\), \(b = -6\), and \(c = -16\). Recognizing these correctly allows us to proceed with solving the equation using the quadratic formula.

Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]It makes use of the coefficients \(a\), \(b\), and \(c\). Here's a step-by-step on how to apply the formula:

• Identify the coefficients \(a\), \(b\), and \(c\).
• Plug these values into the formula.
• Calculate the discriminant, \(b^2 - 4ac\), to evaluate the square root part.
• Solve for \(x\) by doing both the addition and subtraction in the numerator \((\pm)\).

For example, in the equation \(x^2 - 6x - 16 = 0\), by substituting the coefficients into the quadratic formula, we get the solutions \(x = 8\) and \(x = -2\). This method is foolproof for any solvable quadratic equation.

Discriminant

The discriminant of a quadratic equation is a critical component of the quadratic formula. It reveals the nature of the roots of the equation. The discriminant is given by the expression:\( \Delta = b^2 - 4ac \)Here’s how to interpret the discriminant:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), the equation has exactly one real root, which is a repeated root.
• If \(\Delta < 0\), there are no real roots, but two complex roots.

In the problem at hand, the discriminant is calculated as \(100\), which is greater than zero. This indicates that there are two distinct real roots for the equation \(x^2 - 6x - 16 = 0\). Understanding the discriminant's value helps predict the nature of the solutions without solving the entire equation.