Problem 19 - Algebra for College Students

Question

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$x^{2}-18 x+80=0$$

Step-by-Step Solution

Answer

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

Step 1: Identify the coefficients

Firstly, identify the coefficients from the quadratic equation given in the form \[ax^2 + bx + c = 0\] For the equation \(x^2 - 18x + 80 = 0\), the coefficients are: - \(a = 1\) - \(b = -18\) - \(c = 80\).

Step 2: Apply the quadratic formula

The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substitute the values of \(a\), \(b\), and \(c\) into the formula: \[x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot 80}}{2 \cdot 1}\] Simplify inside the square root first.

Step 3: Calculate the discriminant

The discriminant \(b^2 - 4ac\) determines the nature of the roots.For the given equation:\[(-18)^2 - 4 \cdot 1 \cdot 80 = 324 - 320 = 4\]Since the discriminant is positive, there are two distinct real solutions.

Step 4: Evaluate the roots

Now use the discriminant in the quadratic formula: \[x = \frac{18 \pm \sqrt{4}}{2}\] Since \(\sqrt{4} = 2\), substitute back into the formula to get: \[x = \frac{18 \pm 2}{2}\] Thus calculate the two possible values for \(x\).

Step 5: Find the values of x

Calculate each value:1. \[x = \frac{18 + 2}{2} = \frac{20}{2} = 10\]2. \[x = \frac{18 - 2}{2} = \frac{16}{2} = 8\]The solutions to the equation are \(x = 10\) and \(x = 8\).

Step 6: Verify using sum and product relationships

Check the solutions using the sum and product of roots:1. The sum of the roots \(10 + 8 = 18\) matches \(-b/a\), which is \(-(-18)/1 = 18\).2. The product of the roots \(10 \times 8 = 80\) matches \(c/a\), which is \(80/1 = 80\).Both checks confirm the solutions are correct.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation, which takes the general form \(ax^2 + bx + c = 0\). To solve for \(x\), you use the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula directly provides the values of \(x\) that satisfy the equation. The beauty of the quadratic formula is that it works for any quadratic equation, whether the roots are real or complex numbers.
To apply this, start by identifying the coefficients \(a\), \(b\), and \(c\) from your quadratic equation. For example, in the equation \(x^2 - 18x + 80 = 0\), the coefficients are \(a = 1\), \(b = -18\), and \(c = 80\). Substituting these values into the formula allows you to solve for the roots by simplifying step-by-step.
Remember, the calculation involves two operations: addition and subtraction before different results are divided by \(2a\). So, always perform these steps carefully to ensure accuracy.

Discriminant

The discriminant is a key component of the quadratic formula, defined as the expression \(b^2 - 4ac\). It provides valuable information about the nature of the roots of a quadratic equation.

• If the discriminant is positive, you get two distinct real roots.
• If it's zero, there's exactly one real root (a repeated root).
• If the discriminant is negative, the roots are complex or imaginary.

For the equation \(x^2 - 18x + 80 = 0\), the discriminant is calculated as follows: \((-18)^2 - 4 \cdot 1 \cdot 80 = 324 - 320 = 4\). Since the discriminant is 4, which is positive, this means that the equation has two distinct real roots.
Understanding the discriminant not only helps predict the type of roots but also guides whether further simplifications will lead you to real, easy-to-handle numbers or require additional steps for complex solutions.

Sum and Product of Roots

The sum and product of roots provide a simple verification tool for checking your solutions to a quadratic equation. According to Viète's formulas:

• The sum of the roots, \(\alpha + \beta\), should equal \(-\frac{b}{a}\).
• The product of the roots, \(\alpha \cdot \beta\), should equal \(\frac{c}{a}\).

For our specific equation \(x^2 - 18x + 80 = 0\), the

• Sum of the roots is \(10 + 8 = 18\), which matches \(-\frac{-18}{1} = 18\).
• Product of the roots is \(10 \times 8 = 80\), which matches \(\frac{80}{1} = 80\).

This agreement between the calculated and expected values confirms that the roots are correct. Using the sum and product relationships can be an effective method to quickly verify your solutions in terms of correctness and completeness. Even if calculations appear correct through the quadratic formula, confirming with these patterns reinforces the solution's accuracy.