Question

Solve the equation by completing the square. $$x^{2}+16 x=17$$

Short answer

The solutions to the equation are \(x = 1\) and \(x = -17\)

Step-by-step solution

Rewriting The Equation And Adding \((b/2)^{2}\)

Rewrite the equation as \(x^{2}+16x-17=0\). Then, take half of 16, square it and add to both sides of the equation: \(x^{2} +16x+(16/2)^{2} = 17+(16/2)^{2}\). This simplifies to: \(x^{2}+16x+64=81\)

Simplify And Solve For x

The left side becomes a perfect square so it becomes \((x+8)^2 = 81\). So, \(x+8 = \pm \sqrt{81}\) which simplifies to \(x+8=\pm9\)

Solve For x

For \(x+8=9\), subtracting 8 from both sides gives: \(x=1\). For \(x+8=-9\), subtracting 8 from both sides gives: \(x=-17\)

Key concepts

Algebraic Equations

An algebraic equation is like a scale, where each side must be balanced with an equivalent value. These equations can range from simple linear forms like \(x + 2 = 5\) to more complex quadratic forms such as \(x^2 + 16x = 17\) which you encountered in your exercise. The goal when solving an algebraic equation is to isolate the variable, often denoted as 'x', to find the value or values that make the equation true.

When dealing with more complex equations, strategies like 'completing the square' come in handy. This method involves manipulating the arrangement of the terms to create a 'perfect square trinomial', which then can be solved more easily. By adding the square of half the coefficient of 'x' to both sides, we alter the equation to form a square on one side and a constant value on the other, allowing us to isolate 'x' after extracting the square root.

Quadratic Equations

Quadratic equations are a type of polynomial equation that specifically involve an \(x^2\) term and can often be spotted by their standard form \(ax^2 + bx + c = 0\). In your exercise, the quadratic equation \(x^2+16x=17\) had to be solved for 'x'.

To solve these, one can factorize, use the quadratic formula, or complete the square. When factoring is difficult or tedious and you're seeking an alternative to the quadratic formula, completing the square offers a systematic approach for transforming the quadratic equation into a format that readily reveals the value of 'x'. By rearranging the equation to create a perfect square on one side, the solution often becomes more apparent after applying the square root to both sides of the equation.

Perfect Square Trinomials

A perfect square trinomial is a special kind of quadratic expression that results from squaring a binomial. It takes the general form \(a^2 + 2ab + b^2\), which factors back into \((a + b)^2\). The exercise you worked on involved creating a perfect square trinomial by adding \((16/2)^2\) or 64 to both sides of the equation.

In the context of completing the square, forming this trinomial enables you to simplify the equation significantly. Once you've created a perfect square trinomial, the left side of your equation forms a neat square which equates to a solitary number after being simplified, making it possible to solve the original quadratic equation with ease. Not only does this method streamline the process of finding solutions, but it also provides an insightful look at the nature of quadratics and the symmetrical properties they often reveal.