Problem 7 - Algebra A Combined Function

Question

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ x^{2}+7 x+4=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{-7 + \sqrt{33}}{2}\) and \(x = \frac{-7 - \sqrt{33}}{2}\).

Step 1: Identify coefficients

In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). For the equation \(x^2 + 7x + 4 = 0\), we have \(a = 1\), \(b = 7\), and \(c = 4\).

Step 2: Set up the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). We will substitute \(a = 1\), \(b = 7\), and \(c = 4\) into this formula.

Step 3: Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\). This is \(7^2 - 4 \cdot 1 \cdot 4 = 49 - 16 = 33\).

Step 4: Plug values into the quadratic formula

Substitute the values \(b = 7\), \(a = 1\), and the discriminant \(33\) into the formula: \(x = \frac{-7 \pm \sqrt{33}}{2 \cdot 1}\).

Step 5: Simplify the solution

Simplify the expression to solve for \(x\): \(x = \frac{-7 + \sqrt{33}}{2}\) and \(x = \frac{-7 - \sqrt{33}}{2}\). Thus, the solutions are \(x = \frac{-7 + \sqrt{33}}{2}\) and \(x = \frac{-7 - \sqrt{33}}{2}\).

Key Concepts

Discriminant

The discriminant is a key concept in solving quadratic equations. It comes from the quadratic formula, which is: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Here, the term under the square root, \( b^2 - 4ac \), is the discriminant. The value of the discriminant can tell us important information about the types of solutions the quadratic equation will have.

• If the discriminant is positive, as in our equation, the quadratic equation has two distinct real number solutions.
• If it is zero, the quadratic equation has exactly one real number solution, which is sometimes called a repeated or double root.
• If it is negative, there are no real number solutions, and the solutions are complex numbers.

The discriminant helps us understand what kind of roots to expect from our equation without fully solving it.

Real Numbers

Real numbers are an integral part of mathematical concepts and equations. When we talk about real number solutions, we are referring to solutions that are not imaginary and can be located on the number line. An equation like the one we solved,\( x^2 + 7x + 4 = 0 \), has solutions in terms of real numbers because its discriminant was positive.
Real numbers include:

• All positive numbers (like 1, 2, 3.5, etc.).
• All negative numbers (like -1, -2.7, etc.).
• Zero.

Real number solutions mean that these solutions can be exact points on the number line, giving us concrete values rather than abstract ones, such as imaginary or complex numbers. Real roots ensure that solutions are meaningful in real-world applications, like measurements or distances.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) represent coefficients, with \( a eq 0 \) to ensure it remains a quadratic.
All quadratic equations can potentially be solved using the quadratic formula. Our focus equation, \( x^2 + 7x + 4 = 0 \), is a classic example.
Quadratic equations can have:

• Two solutions.
• One solution (when there is a double root).
• Or no real solutions (when the roots are complex).

Being proficient with solving quadratic equations is crucial because they frequently appear in various fields such as physics, engineering, and economics. They help model parabolic relationships and find values in many practical and theoretical problems.