Problem 13 - 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. $$n^{2}+5 n-3=0$$

Step-by-Step Solution

Answer

The solutions are \( n = \frac{-5 \pm \sqrt{37}}{2} \).

Step 1: Identify the coefficients

The given quadratic equation is in the form \(n^2 + 5n - 3 = 0\). Identify the coefficients where \(a = 1\), \(b = 5\), and \(c = -3\).

Step 2: Write the quadratic formula

The quadratic formula is given by \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula will be used to find the solutions of the equation.

Step 3: Substitute coefficients into the formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \( n = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \).

Step 4: Calculate the discriminant

Evaluate the discriminant \(b^2 - 4ac\): \[ 5^2 - 4 \times 1 \times (-3) = 25 + 12 = 37 \].

Step 5: Solve for n using the quadratic formula

Continue solving the quadratic formula:\[ n = \frac{-5 \pm \sqrt{37}}{2} \]. Thus the solutions for \(n\) are \( n = \frac{-5 + \sqrt{37}}{2} \) and \( n = \frac{-5 - \sqrt{37}}{2} \).

Step 6: Check solutions using sum and product relationships

For the quadratic \(n^2 + 5n - 3 = 0\), the sum of roots is \(-b/a = -5/1 = -5\), and the product is \(c/a = -3/1 = -3\). Calculate the sum of solutions:\[ \left(\frac{-5 + \sqrt{37}}{2}\right) + \left(\frac{-5 - \sqrt{37}}{2}\right) = -5 \].Calculate the product of solutions:\[ \left(\frac{-5 + \sqrt{37}}{2}\right) \times \left(\frac{-5 - \sqrt{37}}{2}\right) = \frac{25 - 37}{4} = -3 \]. Both conditions are satisfied.

Key Concepts

Quadratic Formula

Solving quadratic equations can sometimes be tricky, but with the quadratic formula, this task becomes much easier. The quadratic formula allows us to find the roots or solutions of any quadratic equation in the form of \(ax^2 + bx + c = 0\). Here's the secret: the formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). By simply plugging in the coefficients \(a\), \(b\), and \(c\) from the equation, you can find your solutions.

Breaking it down:

• \(-b\) flips the sign of the linear term's coefficient.
• \(\pm\) means two solutions are possible: one by adding the square root term and the other by subtracting it.
• \(\sqrt{b^2 - 4ac}\) is under the square root, known as the "discriminant."
• Everything is divided by \(2a\), the coefficient of the quadratic term doubled.

This formula helps calculate the roots and explicitly demonstrates whether the roots are real or complex based on the term under the square root. For our equation \(n^2 + 5n - 3 = 0\), plugged values give us \(n = \frac{-5 \pm \sqrt{37}}{2}\), leading to two possible solutions for \(n\).

Discriminant

The discriminant is a crucial part of the quadratic formula and tells us a lot about the nature of the roots. It's the part under the square root: \(b^2 - 4ac\). Depending on its value, you'll know instantly whether the solutions are real or complex.

Let's look at the possibilities:

• If the discriminant \(b^2 - 4ac\) is greater than zero, the equation has two distinct real roots.
• If it’s exactly zero, there’s one unique real root (a perfect square).
• If the discriminant is less than zero, brace yourself for complex or imaginary roots.

In our exercise, the discriminant calculation was \(25 + 12 = 37\), which is a positive number. This signals that our equation \(n^2 + 5n - 3 = 0\) indeed has two distinct real roots. Understanding the discriminant not only simplifies calculations but also provides an instant preview of the kind of solutions to expect from the quadratic equation.

Sum and Product of Roots

Besides the quadratic formula, another insightful tool is the relationships involving the sum and product of the roots. After obtaining roots using the formula, you can verify them by calculating their sum and product, which follow specific relations derived from the equation.

These are the rules:

• The sum of the roots \(r_1 + r_2\) is \(-\frac{b}{a}\).
• The product of the roots \(r_1 \cdot r_2\) equals \(\frac{c}{a}\).

For the equation \(n^2 + 5n - 3 = 0\), the sum of the roots calculated is \(-5\) and the product is \(-3\). These checks confirm that our earlier solutions from the quadratic formula were correctly computed, as the sum and product of \(\left(\frac{-5 + \sqrt{37}}{2}\right)\) and \(\left(\frac{-5 - \sqrt{37}}{2}\right)\) satisfy these conditions. Utilizing these relationships gives additional confidence in the results and further deepens the understanding of quadratic equations.