Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$5 x^{2}=500$$

Short answer

The solutions to the equation are \(x = 10\) and \(x = -10\).

Step-by-step solution

Divide by 5

To isolate the term with the variable, divide both sides of the equation by 5: \(x^2 = 100\).

Calculate the Square Root

As we are dealing with a square, take the square root of both sides of the equation to solve for x. Remember that there are two possible square roots, one positive and one negative: \(x = \sqrt{100}\) or \(x = -\sqrt{100}\).

Express the Final Result

In this case, the square root of 100 is an integer (10), so the solution can be simplified to \(x = 10\) or \(x = -10\),

Key concepts

Isolate Variables

When solving quadratic equations, the key first step is often to isolate the variable. This means you need to manipulate the equation in such a way that the variable is on one side by itself. In our textbook example, we have the quadratic equation \(5x^2 = 500\). To isolate the variable, which is \(x\), you divide both sides of the equation by 5 to have \(x^2 = 100\).

Isolating variables makes it possible to apply further methods to solve quadratic equations. Without isolating the variable as the first step, it would be challenging to find the correct solution.

Square Root Method

The square root method is particularly handy when you have a quadratic equation in the form \(x^2 = a\), where \(a\) is some number. After isolating the variable, as explained in the previous section, you can take the square root of both sides to solve for \(x\).

Using our example, after simplifying to \(x^2 = 100\), we apply the square root to both sides resulting in \(x = \sqrt{100}\) or \(x = -\sqrt{100}\). It's crucial to remember to include the positive and negative solutions because squaring either a positive or negative number will give the positive result on the left side of the equation (\(x^2 = a\)).

Radical Expressions

In mathematics, a radical expression is an expression that includes a square root, cube root, or any higher-order root. When you cannot simplify the square root to an integer (as in our case where \(\sqrt{100} = 10\)), the expression stays under the radical symbol. Radical expressions can appear in the solutions of quadratic equations when the squared term does not have a perfect square as in our example.

If the equation had been \(5x^2 = 50\), which simplifies to \(x^2 = 10\), the solution would be radical expressions because \(\sqrt{10}\) is an irrational number and can't be simplified into a whole number. Thus the solutions would be written as \(x = \sqrt{10}\) and \(x = -\sqrt{10}\).

Equation Solutions

The solutions of an equation are the values that we find for the variables that make the equation true. For quadratic equations, this usually means finding the value(s) of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).

In our exercise, we determined that \(x = 10\) and \(x = -10\) are the solution to the equation \(5x^2 = 500\) after simplifying and applying the square root method. A quadratic equation can have at most two real solutions, as seen in this example. These solutions are where the equation crosses the x-axis on a graph, also known as the roots of the equation.