Problem 30 - 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. $$-3 n^{2}-11 n+4=0$$

Step-by-Step Solution

Answer

The solutions are \(n = -4\) and \(n = \frac{1}{3}\). Verified by sum and product.

Step 1: Identify coefficients

The quadratic equation given is \(-3n^2 - 11n + 4 = 0\). Identify the coefficients \(a\), \(b\), and \(c\) where \(a = -3\), \(b = -11\), and \(c = 4\).

Step 2: Write the quadratic formula

The quadratic formula is \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Use this formula to find the solutions for \(n\).

Step 3: Calculate the discriminant

Substitute \(a\), \(b\), and \(c\) into the discriminant part of the formula: \(b^2 - 4ac\). Calculate \((-11)^2 - 4(-3)(4) = 121 + 48 = 169\).

Step 4: Find the solutions

Using the discriminant, substitute into the quadratic formula: \( n = \frac{-(-11) \pm \sqrt{169}}{2 \times -3} \). This becomes \( n = \frac{11 \pm 13}{-6} \). Solve each part: \( n = \frac{11 + 13}{-6} = -4 \) and \( n = \frac{11 - 13}{-6} = \frac{1}{3} \).

Step 5: Verify solutions with sum and product

The sum of the roots \(-\frac{b}{a} = \frac{-(-11)}{-3} = \frac{11}{-3}\) must equal the sum of \(-4\) and \(\frac{1}{3}\). Converting to a common denominator: \(-4 = \frac{-12}{3}\), so the sum \(\frac{-11}{3}\) matches. The product \(\frac{c}{a} = \frac{4}{-3}\) is the same as \(-4 \times \frac{1}{3} = \frac{-4}{3}\). Both conditions are met.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides a systematic way to find the roots, or solutions, of these equations. The formula is given by:

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

This formula might look complex, but it's actually quite straightforward once we understand its components. Here's a breakdown of each part:

• \( -b \) represents the negation of the linear coefficient.
• \( \pm \) indicates that there could be two solutions, due to the square root operation.
• \( \sqrt{b^2 - 4ac} \) is called the discriminant, which we will explore further in another section.
• \( 2a \) acts as the divisor for the whole expression, scaling the result.

By substituting the values of \( a \), \( b \), and \( c \) from the quadratic equation into this formula, we can find the exact values that make the equation zero.

Sum and Product of Roots

The sum and product of the roots of a quadratic equation are intriguing properties that can provide quick checks for our solutions. For an equation \( ax^2 + bx + c = 0 \), these relationships are:

• Sum of the roots \( = -\frac{b}{a} \)
• Product of the roots \( = \frac{c}{a} \)

Let's see how these relationships work:

• By calculating \( -\frac{b}{a} \), we find the sum of the two roots. This is useful in verifying our solutions from the quadratic formula. If our roots are correct, their sum should match this expression.
• Similarly, \( \frac{c}{a} \) checks the product of the roots, providing another way to ensure the solutions are accurate.

These shortcuts are based on the form of the quadratic equation itself and reflect a deeper symmetry in its structure.

Discriminant in Quadratic Equations

The discriminant is a vital part of the quadratic formula that helps determine the nature of the roots. It is given by the formula:

• \( \Delta = b^2 - 4ac \)

Here's why the discriminant is so essential:

• It indicates the number and type of roots a quadratic equation possesses.
• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, or a repeated root.
• If \( \Delta < 0 \), the roots are complex and not real.

Understanding the discriminant allows us to predict whether to expect real solutions or if we need to look for complex ones before even solving the equation. This can save time and guide us appropriately in mathematical explorations.