Problem 23 - Elementary Algebra

Question

Solve each quadratic equation using quadratic formula. $$ (x+1)(x+4)=6 $$

Step-by-Step Solution

Answer

Answer: The solutions to the quadratic equation are \(x_1 = \frac{-5 + \sqrt{33}}{2}\) and \(x_2 = \frac{-5 - \sqrt{33}}{2}\).

Step 1: Rewrite the equation in standard quadratic form

To rewrite the equation in standard quadratic form, we need to expand the given expression and subtract 6 from both sides: $$ (x+1)(x+4) = 6 \\ x^2 + 4x + x + 4 = 6 \\ x^2 + 5x + 4 = 6 \\ x^2 + 5x - 2 = 0 $$ Now, the equation is in standard quadratic form, with \(a=1\), \(b=5\), and \(c=-2\).

Step 2: Apply the quadratic formula

The quadratic formula is given by: $$ x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a} $$ We need to plug in the values of \(a\), \(b\), and \(c\) that we found in step 1: $$ x = \frac{-5 \pm \sqrt{5^2 -4(1)(-2)}}{2(1)} $$

Step 3: Simplify the expression

Calculate what's inside the square root and simplify the expression: $$ x = \frac{-5 \pm \sqrt{25 + 8}}{2} \\ x = \frac{-5 \pm \sqrt{33}}{2} $$

Step 4: Write the final solutions

There are two solutions, one with a plus sign and one with a minus sign in the quadratic formula: $$ x_1 = \frac{-5 + \sqrt{33}}{2} \\ x_2 = \frac{-5 - \sqrt{33}}{2} $$ The quadratic equation \((x+1)(x+4)=6\) has two solutions: \(x_1 = \frac{-5 + \sqrt{33}}{2}\) and \(x_2 = \frac{-5 - \sqrt{33}}{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool in solving quadratic equations, which are equations in which the highest power of the variable is 2. Whenever we face such problems and cannot easily factorize the expression or complete the square, the quadratic formula becomes our go-to method. It is expressed as:

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

This shows how versatile the formula can be to find solutions for any quadratic expression. Here, the letters \(a\), \(b\), and \(c\) are coefficients derived from the standard quadratic form of the equation. By substituting the corresponding coefficients into this formula, you can find potential solutions for \(x\), known as roots.

Standard Quadratic Form

The standard quadratic form is a way of organizing quadratic equations to make processes like solving easier. This form is expressed as \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are numbers, with \(a eq 0\). Before applying the quadratic formula, an equation often needs to be converted into this form.

For instance, if given \((x+1)(x+4)=6\), it initially represents a product. By expanding and simplifying, you eventually reform it into the standard quadratic form, \(x^2 + 5x - 2 = 0\). This critical restructuring isolates all terms to one side: the quadratic expression, the linear term, and the constant term, making the problem ready for applying several solving techniques.

Solving Equations

Solving equations involves finding the values of the unknowns that satisfy them. Quadratic equations specifically result in two possible values for the unknown variable, expressed as \(x\), which is found using various methods like the quadratic formula, factoring, or completing the square.

• Identify: Recognize the type of equation you are dealing with, such as a quadratic equation in this context.
• Reformulate: Ensure the equation is in the standard quadratic form.
• Apply: Use an appropriate method to find the values of \(x\). In problems involving quadratic equations, using the quadratic formula ensures solutions regardless of factorability.
• Verify: Always substitute your found values back into the original equation to confirm they are correct.

Math Problem Solving

Math problem solving with quadratic equations involves logical steps and thorough checking. It often requires rearranging equations to a standard, understandable form, employing appropriate strategies like the quadratic formula, and verifying outcomes. This approach is systematic and helps solve more complex problems that might not have immediate or simple solutions.

Critical thinking is essential in math problem solving, which includes:

• Analyzing: Understand what the problem is asking and identify necessary information.
• Simplifying: Break down the problem into smaller, manageable parts, like rewriting equations.
• Calculating: Carry out the calculations with care, especially when involving square roots, and combining terms.
• Revising: Always recheck each step to prevent errors, ensuring that your solutions do solve the original problem.

Ultimately, developing a strategy that works with different types of quadratic equations enhances your math problem-solving skills overall.