Problem 67 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. (Lesson 9.6) $$5=6 x^{2}+7 x$$

Step-by-Step Solution

Answer

The solutions for the given quadratic equation are \(x_{1} = -1/2 \) and \(x_{2} = -5/3 \)

Step 1: Rewrite the equation

In order to solve the quadratic equation, it should be written in the format \(ax^{2} + bx + c = 0\). In our case, this equation can be rearranged as: \(6x^{2} + 7x - 5 = 0\)

Step 2: Identify the coefficients

Next, recognize the values of a, b, and c. From the rearranged equation it can be observed that: \(a = 6\), \(b = 7\), and \(c = -5\)

Step 3: Substitute the coefficients in the quadratic formula

Put the identified coefficients in the quadratic formula and calculate the roots: For \(x_{1}\): \(x_{1} = [-7 + \sqrt{(7)^{2}-4*6*(-5)}] / (2*6) = -1/2 \)For \(x_{2}\): \(x_{2} = [-7 - \sqrt{(7)^{2}-4*6*(-5)}] / (2*6) = -5/3 \)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It allows us to find the roots of any quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \). The formula states:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is derived from the process of completing the square and is applicable when the equation has real or complex solutions.
Use the quadratic formula when factoring does not readily simplify the equation or when the equation doesn't easily lend itself to other methods. It provides two solutions, \( x_1 \) and \( x_2 \), where the "±" symbol indicates we calculate two variations: one adding and one subtracting the square root.
To use it correctly, remember:

• Identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation.
• Substitute these values into the formula and compute to find the roots.

Solving Quadratic Equations

Solving quadratic equations can be approached in several ways. Common methods include:

• Factoring
• Using the quadratic formula
• Completing the square
• Graphing

For our example \(6x^2 + 7x - 5 = 0\), using the quadratic formula is most effective.
Begin by ensuring the equation is correctly formatted, like \( ax^2 + bx + c = 0 \). This step guarantees zero is on one side of the equation, optimizing conditions for finding solutions. Next, identify the coefficients and substituting them into the quadratic formula.
It's important to note that different equations may suit different methods, so understanding all solving techniques provides flexibility and strength when tackling various problems.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them more workable.
For quadratic equations, it's crucial to set the equation to a standard form before solving. By moving all terms to one side of the equation, you set it against zero: \( ax^2 + bx + c = 0 \).
In our example, it involved:

• Subtracting 5 from both sides of the initial equation \(5 = 6x^2 + 7x\).
• This became \(6x^2 + 7x - 5 = 0\).

This straightforward step is fundamental for solving because it allows the use of methods like the quadratic formula effectively.
Algebraic manipulation also involves recognizing opportunities to simplify expressions, factor where applicable, and balance equations efficiently to solve them quickly and with clarity.