Problem 59 - Elementary Algebra

Question

Solve. $$5 x 2+5 x+4+x+1 x 2+3 x-4=5 x 2-1$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{{-9 + \sqrt{77}}}{2}\) and \(x = \frac{{-9 - \sqrt{77}}}{2}\).

Step 1: Simplify both sides of the equation

First, we combine like terms on both sides of the equation. On the left side, combine terms: \(5x^2 + 1x^2\) to get \(6x^2\), and \(5x + x + 3x\) to get \(9x\). Simplifying further, the left side becomes \(6x^2 + 9x + 0\) after cancelling \(+4\) and \(-4\). The right side remains \(5x^2 - 1\). The equation is now: \[6x^2 + 9x = 5x^2 - 1\].

Step 2: Move all terms to one side

To solve for \(x\), we should have a polynomial equation equal to zero. Subtract \(5x^2\) from both sides of the equation to get: \[6x^2 - 5x^2 + 9x = -1\]. Simplify to: \[x^2 + 9x = -1\]. Then, add 1 to both sides to move the constant term as well. This results in: \[x^2 + 9x + 1 = 0\].

Step 3: Solve the quadratic equation

The equation \(x^2 + 9x + 1 = 0\) is a standard quadratic form, \(ax^2 + bx + c = 0\). We will use the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). For our equation, \(a = 1\), \(b = 9\), \(c = 1\).

Step 4: Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\): \[9^2 - 4 \times 1 \times 1 = 81 - 4 = 77\].

Step 5: Compute the roots

Since the discriminant is positive, the equation has two real roots. Use the quadratic formula to find them: \[x = \frac{{-9 \pm \sqrt{77}}}{2}\]. So, the solutions are \[x = \frac{{-9 + \sqrt{77}}}{2}\] and \[x = \frac{{-9 - \sqrt{77}}}{2}\].

Key Concepts

Combining Like Terms

When solving equations, especially quadratic ones, the process often begins with combining like terms. **Why is this important?** It simplifies the equation, making it easier to manipulate and ultimately solve. To combine like terms, look for terms with the same variable and exponent:

• Coefficients: These are numbers that multiply the variables, like the 5 in \(5x^2\).
• Like Terms: Terms with the same variable and exponent, such as \(5x^2\) and \(1x^2\), can be combined.

In our equation, we combined

• \(5x^2 + 1x^2\) to become \(6x^2\).
• Similarly, \(5x + x + 3x\) was combined to become \(9x\).

By doing this, the equation is simplified and ready for the next solving steps.

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation in the form \(ax^2 + bx + c = 0\). **What is the formula?** It is expressed as: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]**Steps to use the quadratic formula:**1. Identify \(a\), \(b\), and \(c\) in your equation. Here, they were 1, 9, and 1 respectively.2. Substitute these values into the formula.3. Solve for \(x\) by evaluating the expression.The formula allows finding the roots of the equation, where it intersects the x-axis. It's useful when factoring is complex or impossible.

Discriminant

The discriminant is a concept within the quadratic formula, represented by \(b^2 - 4ac\). It provides critical information about the roots of the quadratic equation. **What does it tell us?** Depending on its value:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is exactly one real root (a repeated root).
• If the discriminant is negative, there are no real roots, but two complex roots.

In our solved equation, the discriminant was calculated as 77. Since it is positive, this indicates that the equation has two real and distinct solutions. Calculating the discriminant helps us predict the nature of the solutions before actually solving the equation.