Problem 19 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ x^{2}-18 x+80=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 10 \) and \( x = 8 \).

Step 1: Write Down the Quadratic Formula

The given equation is a quadratic equation, which can be solved using the quadratic formula, given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In our equation, \( a = 1 \), \( b = -18 \), and \( c = 80 \).

Step 2: Compute the Discriminant

Calculate the discriminant, \( \Delta = b^2 - 4ac \). Substituting the values, we get: \( \Delta = (-18)^2 - 4 \cdot 1 \cdot 80 = 324 - 320 = 4 \).

Step 3: Calculate the Roots

Since the discriminant \( \Delta = 4 \) is a positive perfect square, there are two distinct real roots. Substitute into the quadratic formula: \( x = \frac{-(-18) \pm \sqrt{4}}{2 \times 1} \). This simplifies to \( x = \frac{18 \pm 2}{2} \).

Step 4: Simplify Each Solution

Split the expression into two solutions: \( x = \frac{18 + 2}{2} = 10 \) and \( x = \frac{18 - 2}{2} = 8 \).

Step 5: Verify the Solutions

Substitute \( x = 10 \) and \( x = 8 \) back into the original equation \( x^2 - 18x + 80 = 0 \) to confirm both solutions work. For \( x = 10 \), \( 10^2 - 18 \times 10 + 80 = 0 \) and for \( x = 8 \), \( 8^2 - 18 \times 8 + 80 = 0 \) confirm each solution as valid.

Key Concepts

Quadratic Formula

To solve a quadratic equation efficiently, you can use the quadratic formula. This formula provides a reliable method to find the roots of a quadratic equation of the form \( ax^2 + bx + c = 0 \). The quadratic formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula will solve any quadratic equation, regardless of whether it is factorable or not. The values \( a \), \( b \), and \( c \) correspond to the coefficients of the polynomial. Here's what you need to do:

• Identify the coefficients \( a \), \( b \), and \( c \) from your equation.
• Substitute these values into the quadratic formula.
• Calculate the expression to find the values for \( x \).

In the given equation \( x^{2}-18x+80=0 \), we found \( a = 1 \), \( b = -18 \), and \( c = 80 \). This offers a direct pathway to finding the roots.

Discriminant

The discriminant provides crucial information about the nature of the roots of a quadratic equation. It is found in the quadratic formula under the square root: \( b^2 - 4ac \). The value of the discriminant determines the type of roots:

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

In our example, the discriminant was calculated as \( \Delta = 4 \), which is positive and a perfect square. This indicates that we have two distinct real roots. The discriminant not only tells you the nature of the roots but also their type, which can help in visualizing the solutions geometrically on a graph.

Roots of a Quadratic Equation

The roots or solutions of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). Using the quadratic formula, we solve for these roots. Here's a quick recap on how to find them:

• Plug \( b^2 - 4ac \) into the quadratic formula to find the expression under the square root.
• If the discriminant is positive, compute both the positive and negative square roots.
• Substitute these values back into the equation \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find each root.

In our solved equation, the roots were \( x = 10 \) and \( x = 8 \). To verify, these values should satisfy the original equation when substituted back. Understanding the roots not only solves the algebraic equation but allows you to interpret possible intersection points of the corresponding parabola with the x-axis.