Question

What is the value of \(x ?\) (1) \(x^{2}-9=16\) (2) \(3 x(x-5)=0\)

Short answer

The value of \(x\) is 5.

Step-by-step solution

Solve Equation (1)

Start with the equation \[ x^2 - 9 = 16 \] Add 9 to both sides to isolate the quadratic term:\[ x^2 = 25 \] Take the square root of both sides to solve for \(x\): \[ x = \pm 5 \] This gives two potential solutions, \(x = 5\) and \(x = -5\).

Solve Equation (2)

Start with the equation \[ 3x(x - 5) = 0 \] Factor the equation into two parts: \[ 3x = 0 \ \, or \, \ x - 5 = 0 \] Solve each individual part:\[ x = 0 \ \, or \, \ x = 5 \] This gives two potential solutions, \(x = 0\) and \(x = 5\).

Find Common Solution

Compare the potential solutions from both equations. From Step 1: \(x = 5\) and \(x = -5\); from Step 2: \(x = 5\) and \(x = 0\). The common solution is \(x = 5\).

Key concepts

quadratic equations

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants. In these equations, \(a e 0\) to ensure it's quadratic. The term \(ax^2\) is what makes it a quadratic equation since square (\text{quadratus} in Latin) refers to the second power. Quadratic equations often appear in standardized tests like GMAT and GRE. They can be solved using various methods like factoring, using the quadratic formula, or completing the square. These methods help find the roots \(x\), which satisfy the equation. For example, consider the equation \(x^{2} - 9 = 16.\) First, adding \(9\) to both sides gives \(x^2 = 25.\) Taking the square root of both sides results in \(x = \pm \sqrt{25} = \pm 5\). Thus, the solutions are \(x = 5\) and \(x = -5\).

solving for x

When we solve an equation for \x, we are finding the value or values of \x that make the equation true. This is also known as finding the roots or solutions of the equation. To solve quadratic equations, follow these general steps:

• Isolate the quadratic term if needed.
• Rewrite the equation in the standard quadratic form \(ax^2 + bx + c = 0\).
• Use factoring, quadratic formula, or other methods to find \x.

For example, from our problem \(3x(x - 5) = 0\), we can factor the equation into two parts to solve for \x. Setting each factor to zero gives us:

• \3x = 0 \rightarrow x = 0\
• \x - 5 = 0 \rightarrow x = 5\

So, the solutions are \x = 0\ and \x = 5\.

factoring equations

Factoring is a key method to solve quadratic equations by expressing them as a product of simpler binomial terms. The factored form of a quadratic can make solving for \x easier. The steps to factor a quadratic equation \(ax^2 + bx + c = 0\) are:

• Find two numbers that multiply to \(ac\) and add to \b\.
• Rewrite \bx\ as the sum of these two numbers.
• Factor by grouping.

The problem \(3x(x - 5) = 0\) is already in its factored form. We simply set each factor to zero to find the solutions. This gives us \(3x = 0\ \Rightarrow x = 0\) and \(x - 5 = 0\ \Rightarrow x = 5\). Thus, our solutions are \x = 0\ and \x = 5\.

GRE math review

The GRE tests your ability to solve algebra problems, including quadratic equations. Understanding quadratic equations and factoring is crucial for doing well.During the GRE, you might encounter problems similar to the one discussed. Knowing how to isolate \x\, factor the equation, and find common solutions will save you time and effort.To summarize the process:

• Start by isolating the quadratic term.
• Rewrite the equation in its standard form.
• Use appropriate methods to find \x\.
• Compare potential solutions if dealing with multiple equations.

In our example \x = 5\ was the common solution for both equations \(x^{2} - 9 = 16\) and \(3x(x - 5) = 0.\) Remember, practice is key to mastering these concepts and excelling in the GRE math section.