Problem 10 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$x^{2}-3 x-10=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^2 - 3x - 10 = 0\) are \(x = 5\) and \(x = -2\).

Step 1: Identify the coefficients

The equation \(x^2 - 3x - 10 = 0\) is already in the standard quadratic form \(ax^2 + bx + c = 0\). So we can identify the coefficients as \(a = 1\), \(b = -3\), and \(c = -10\).

Step 2: Apply the quadratic formula

By substituting \(a\), \(b\), and \(c\) into the quadratic formula, we have \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4*1*(-10)}}{2*1}\), which simplifies to \(x = \frac{3 \pm \sqrt{9 + 40}}{2}\).

Step 3: Simplify the roots

Simplify the square root to get \(x = \frac{3 \pm \sqrt{49}}{2}\). So, \(x = \frac{3 \pm 7}{2}\). That means we have two solutions, \(x = \frac{3 + 7}{2} = 5\) and \(x = \frac{3 - 7}{2} = -2\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are an essential part of algebra, often encountered by students across various levels of education. Solving these equations means finding the values of the variable, commonly denoted as 'x', which make the equation true. A quadratic equation takes the general form of ax^2 + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'a' is not zero.

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a powerful tool that can solve any quadratic equation, even when factoring is not possible. It directly provides the roots of the equation by substituting the coefficients 'a', 'b', and 'c' from the quadratic equation into the formula. This method is universally applicable and ensures that learners can handle even the most challenging quadratic equations.

Quadratic Equation roots

When using the quadratic formula to solve for the roots of a quadratic equation, we are essentially looking for the points where the quadratic graph intersects the x-axis. These points, called roots or zeros, are the solutions to the equation. There are typically two roots for a quadratic equation, which can be real or complex.

The discriminant, \( D = b^2 - 4ac \), plays a vital role in determining the nature of the roots. If it is positive, the equation has two distinct real roots. If it is zero, there is exactly one real root, known as a repeated root. However, if the discriminant is negative, the equation has two complex roots. In our example, \( x^2 - 3x - 10 = 0 \), the discriminant is positive (\(\sqrt{49}\)), leading to two distinct real solutions, 5 and -2.

Simplifying Irrational Solutions

Sometimes, applying the quadratic formula leads to solutions that involve irrational numbers, which can be intimidating for learners. An irrational solution arises when the discriminant is not a perfect square, resulting in a square root of a non-square number. The goal is to simplify the expression as much as possible to make it understandable.

To simplify, look for the largest square factor of the number within the radical, and express it as the product of two square roots. Then, take the square root of the perfect square out of the radical, leaving the non-square part inside. For instance, if we encountered a solution like \(x = \frac{3 \pm \sqrt{72}}{2}\), we would simplify \(\sqrt{72}\) as \(\sqrt{36*2}\), which becomes \(6\sqrt{2}\). Finally, our simplified solution would be \(x = \frac{3 \pm 6\sqrt{2}}{2}\). In the given example, however, the solutions are rational numbers, implying that no further simplification of the roots is necessary.