Question

Determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation. \(x^2-16 x+64=0\)

Short answer

You can solve this quadratic equation by factoring. The equation represents a perfect square trinomial, so it can be factored as \((x-8)^2\). By solving for 'x' from this factor, you get \(x = 8\).

Step-by-step solution

Identification

Identify the given quadratic equation as a perfect square trinomial. \(x^2 - 16x + 64 = 0\) can be written as \((x - 8)^2 = 0\), which is a perfect square trinomial.

Factoring the perfect square trinomial

Factor the perfect square trinomial using the formula \((a - b)^2 = a^2 - 2ab + b^2\). So in this case, \(a = x\) and \(b = 8\). Therefore, factoring gives us \((x - 8)^2\).

Solving for the variable

To solve for 'x', first remove the square on \(x - 8\) by taking the square root on both sides. This gives us \(x - 8 = 0\) or \(x - 8 = -0\) (since squaring gives both positive and negative results). Solving for 'x' in \(x - 8 = 0\) gives \(x = 8\). As squaring resulted only in positive number, we discard \(x - 8 = -0\). So, the solution is \(x = 8\).

Key concepts

Factoring

Factoring is a crucial method for solving quadratic equations. It involves rewriting a quadratic equation as a product of two binomials, if possible. This technique simplifies the equation, making it easier to solve. Here's how it works:

• Start with a quadratic equation in the form of \(ax^2 + bx + c = 0\).
• Try to express it as \((dx + e)(fx + g) = 0\).

Once factored, each binomial can be solved separately to find the values of \(x\). This method is especially useful when the quadratic is a "nice" equation, where the constants are simple to work with. In our exercise, the equation \(x^2 - 16x + 64 = 0\) was factored immediately into a single squared binomial, \((x - 8)^2\), due to its perfect square trinomial nature.

Perfect Square Trinomials

Perfect square trinomials are special quadratic expressions that take the form \((a-b)^2 = a^2 - 2ab + b^2\). Recognizing these can greatly simplify solving quadratics. By seeing the structure, you might quickly rewrite it as the square of a binomial.

• Check if the first and last terms are perfect squares.
• Consider if twice the product of their roots equals the middle term.

For the equation \(x^2 - 16x + 64 = 0\), this pattern is visible. Here, \(x^2\) and \(64\) are perfect squares with roots \(x\) and \(8\) respectively, and their twice product is \(16x\). So it becomes \((x - 8)^2\), simplifying further solving.

Solving Quadratic Equations

Solving quadratic equations can be done using several methods: factoring, completing the square, or using the quadratic formula. Let's focus on solving using perfect square trinomials, as seen in our exercise.

• Identify if the quadratic expression is a perfect square.
• Rewrite the equation using the squared binomial form.
• Set this equal to zero \((a-b)^2 = 0\).
• Take the square root of both sides, solving for the variable \(x\).

In our given example, once rewritten as \((x - 8)^2 = 0\), taking square roots yields \(x - 8 = 0\). Solving, we find \(x = 8\). This method quickly gives the solution by using properties of squares, making it efficient when applicable.