Question

If \(x^2-5 x=24\), which of the following is a possible value of \(x\) ? A) \(-8\) B) 3 C) 8 D) 24

Short answer

Option C. \(x = 8\).

Step-by-step solution

Rewrite the equation in the form of a quadratic equation

First, rewrite the equation as a quadratic equation: \(x^2 - 5x - 24 = 0\).

Solve for x using factoring, completing the square, or quadratic formula

We can attempt to factor this quadratic equation to find the solutions of x. Otherwise, we would need to use the quadratic formula. Factoring the quadratic equation: \((x - 8)(x + 3) = 0\)

Identify the possible values of x

Now, set each factor equal to zero and solve for \(x\): \(x - 8 = 0 \Rightarrow x = 8\), \(x + 3 = 0 \Rightarrow x = -3\).

Compare the possible values with the given options

We have found that \(x\) could be either 8 or -3. Now, we'll compare these values with the options available: A) -8 (Not a solution we found) B) 3 (Not a solution we found) C) 8 (One of the solutions we found) D) 24 (Not a solution we found) Therefore, the correct answer is: Option C. \(x = 8\).

Key concepts

Factoring Quadratic Equations

Factoring quadratic equations is a method used to solve equations of the form \(ax^2 + bx + c = 0\). This technique involves finding two numbers that multiply to the constant term, \(c\), and add up to the linear coefficient, \(b\).
For example, when solving \(x^2 - 5x - 24 = 0\), we factor it as \((x - 8)(x + 3) = 0\).

• The numbers -8 and 3 multiply to -24 (which is \(c\)) and add up to -5 (which is \(b\)).
• Once the equation is factored, we can easily find the solutions by setting each factor equal to zero.

This method is straightforward and effective when the quadratic can be factored easily.
However, not all quadratic equations are factorable using integers, which is when alternative methods like the quadratic formula or completing the square become useful.

Solving Quadratic Equations

Solving quadratic equations means finding the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).
There are several methods to achieve this, including:

• Factoring
• Completing the square
• Using the quadratic formula

Each method has its own pros and cons.

• Factoring is quick when applicable, but it only works if the quadratic can be neatly factored into integer solutions.
• Completing the square is a systematic method that can solve any quadratic equation but may be time-consuming.
• The quadratic formula is a go-to method that always works, applicable to any quadratic equation.

To solve a given quadratic, the choice of method often depends on the specific values of \(a\), \(b\), and \(c\), and sometimes on personal preference or convenience.

Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations. It can solve any equation of the form \(ax^2 + bx + c = 0\) and is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows us to find the values of \(x\) by substituting \(a\), \(b\), and \(c\) into the equation.

• First, calculate the discriminant, \(b^2 - 4ac\).
• If the discriminant is positive, there are two distinct real solutions. If it is zero, there is one real solution. If negative, the solutions are complex.

For the quadratic \(x^2 - 5x - 24 = 0\), by plugging in \(a = 1\), \(b = -5\), and \(c = -24\), you can confirm the solutions \(x = 8\) and \(x = -3\), showing this formula’s effectiveness regardless of whether factoring is feasible.

Completing the Square

Completing the square is another technique to solve any quadratic equation. This involves rearranging the equation into a perfect square trinomial, which can be solved by taking the square root of both sides.
Here's how the process works:

• Begin with the equation in the form \(ax^2 + bx = -c\).
• Divide the entire equation by \(a\) (if \(aeq 1\)) to make the leading coefficient 1.
• Add and subtract \((b/2a)^2\) inside the equation to form a perfect square trinomial.
• Rewrite the equation as \((x + d)^2 = e\) and solve for \(x\) by taking the square root of both sides.

Using this method for \(x^2 - 5x - 24 = 0\) confirms that \(x = 8\) or \(x = -3\), aligning with solutions obtained through factoring.
Completing the square is particularly useful when solving quadratics that do not factor easily.