Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-48: Use the quadratic formula to factor the expressions. $$ x^{2}+x-30 $$

Step-by-Step Solution

Answer

The factored form of the expression is \((x - 5)(x + 6)\).

Step 1: Identify coefficients

The quadratic expression \(x^2 + x - 30\) is in the form \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 1\), and \(c = -30\).

Step 2: Write down the quadratic formula

The quadratic formula is used to find the roots of any quadratic equation and is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Step 3: Calculate the discriminant

Substitute the coefficients into the formula to find the discriminant: \( b^2 - 4ac = 1^2 - 4 imes 1 imes (-30) = 1 + 120 = 121 \).

Step 4: Calculate the square root of the discriminant

The square root of the discriminant is \( \sqrt{121} = 11 \).

Step 5: Find the roots using the quadratic formula

Substitute the square root of the discriminant and the coefficients into the quadratic formula:\[ x = \frac{-1 \pm 11}{2} \]. This gives two potential solutions for \(x\).

Step 6: Solve for the roots

Calculate the two solutions:1. \( x = \frac{-1 + 11}{2} = \frac{10}{2} = 5 \)2. \( x = \frac{-1 - 11}{2} = \frac{-12}{2} = -6 \)

Step 7: Write the factored form using the roots

Since the roots are \(5\) and \(-6\), the factored form of the expression is \( (x - 5)(x + 6) \).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a given expression into a product of two simpler expressions, often in the form \((x - r_1)(x - r_2)\). Here, \(r_1\) and \(r_2\) are the roots or zeros of the quadratic equation. This process reveals the values that, when substituted into the quadratic expression, result in zero.

When faced with a quadratic expression like \(x^2 + x - 30\), factoring can simplify finding solutions and graphing. The goal is to represent it in a way that suggests its roots. This method makes it possible to easily identify the x-intercepts of the corresponding quadratic function.

• Identify coefficients: Recognize\(a\), \(b\), \(c\) in \(ax^2 + bx + c\).
• Find roots: Use the quadratic formula or factoring technique.
• Write the expression: Convert back into the factored form.

By rewriting the expression \(x^2 + x - 30\) into its factored form \((x - 5)(x + 6)\), it's clear that the solutions to \(x^2 + x - 30 = 0\) are \(x = 5\) and \(x = -6\). Factoring is a powerful tool in algebra that simplifies working with quadratics.

Discriminant Calculation

The discriminant is a vital part of the quadratic formula, encased within the square root: \(b^2 - 4ac\). This value determines the nature and number of the roots of a quadratic equation. Calculating the discriminant helps identify whether solutions are real, repeated, or complex without having to solve it fully.

For the quadratic expression \(x^2 + x - 30\), calculating the discriminant involves:

• Using the values \(a = 1\), \(b = 1\), \(c = -30\).
• Computing \(b^2 - 4ac = 1^2 - 4\times 1 \times (-30) = 121\).

Since the discriminant is 121, a positive perfect square, it indicates two distinct real roots. This calculation reveals not just the presence of real solutions, but also hints at their rationality owing to the square root being an integer.
Knowing the role of the discriminant in the solving process provides insight into the behavior of the graph of the quadratic equation, especially the intercepts on the x-axis.

Roots of Quadratic Equations

Roots of a quadratic equation are the solutions for \(x\) that satisfy \(ax^2 + bx + c = 0\). Finding these solutions helps understand the equation's characteristics, such as points where the graph crosses the x-axis.

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) plays a crucial role in identifying these roots. Here's how it applied to \(x^2 + x - 30\):

• With \(a = 1\), \(b = 1\), and \(c = -30\), calculate the discriminant: 121, a perfect square.
• Substitute into formula: \(x = \frac{-1 \pm 11}{2}\).
• Compute the solutions: \(x = 5\) and \(x = -6\).

These roots suggest the quadratic expression touches the x-axis at these points, effectively making them zeros. Understanding how to find and interpret these roots not only solves algebraic challenges but also offers graphical insights into the shape and orientation of parabolas.