Question

Determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation. \(x^2-100=0\)

Short answer

The solutions to the equation \(x^2-100=0\) are \(x=10\) and \(x=-10\).

Step-by-step solution

Use the square root property

Since the equation is in the form of \(x^2=c\), we can directly apply the square root property which says that if \(x^2=c\), then \(x\) is either \(\sqrt{c}\) or \(-\sqrt{c}\). In this case, \(c\) is 100, so the solutions would be \(\sqrt{100}\) or \(-\sqrt{100}\).

Simplify the square roots

The square root of 100 is 10, so the solutions are \(x=10\) and \(x=-10\)

Key concepts

Factoring

Factoring is a technique used to solve quadratic equations by expressing the equation as a product of its factors. This method is quite handy when the quadratic equation is in the form of \(ax^2 + bx + c = 0\).
To use factoring, follow these steps:

• Write the equation in standard form: \(ax^2 + bx + c = 0\).
• Identify two numbers that multiply to give \(ac\) (the product of \(a\) and \(c\)) and add up to \(b\).
• Rewrite the middle term using the two numbers found, and then factor by grouping.
• Set each factor to zero and solve for \(x\).

However, in the given problem, \(x^2 - 100 = 0\) is already in a form that can be solved directly with simpler methods like using square roots. Factoring is generally less straightforward in this case, because there's no middle term (\(b = 0\)), making the equation perfect for the square root method instead.

Square Roots

Solving quadratic equations using square roots is one of the more straightforward methods and works particularly well for equations in the form \(x^2 = c\).
This equation can be directly solved using the square root property, which states:

If \(x^2 = c\), then \(x = \sqrt{c}\) or \(x = -\sqrt{c}\).

This technique is used effectively in the original exercise:

• The given equation is \(x^2 - 100 = 0\).
• Add 100 to both sides: \(x^2 = 100\).
• Apply the square root property: \(x = \sqrt{100}\) or \(x = -\sqrt{100}\).
• The solutions are \(x = 10\) and \(x = -10\), since the square root of 100 is 10.

Using square roots is advantageous here because it's quick and eliminates the need for complex calculations or factoring.

Completing the Square

Completing the square is a useful technique when you want to solve quadratic equations that aren't easily factorable. This method involves turning a quadratic equation into a perfect square trinomial. While not used for the given exercise, it's important to understand how it works:

• Start with the equation in the form \(ax^2 + bx + c = 0\).
• If \(a eq 1\), divide the whole equation by \(a\) to make \(x^2\) have a coefficient of 1.
• Move the constant term, \(c\), to the other side of the equation.
• Calculate \((\frac{b}{2})^2\), and add it to both sides of the equation, making the left-hand side a perfect square trinomial.
• Factor the perfect square trinomial into \((x + p)^2\), and solve for \(x\) by taking the square root of both sides.

While completing the square can be more time-consuming and is unnecessary for simple equations like \(x^2 - 100 = 0\), it's a powerful tool for more complex quadratics or when working with functions in vertex form.