Problem 36 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The solution to given equation are \(x = 1\) and \(x = -17\)

Step 1: Arrange the equation

In our equation \( x^{2} + 16x = 17 \) , we have the terms on one side and constant on another. This is suitable for completing the square process.

Step 2: Complete the square

The number that completes the square is given by \(b/2\) squared where b is the coefficient of x in the quadratic equation. Here 'b' is 16 , so our number is \((16/2)^{2} = 64\). Consider this number and re-write the equation as \(x^{2} + 16x + 64 = 17 + 64\)

Step 3: Simplify the equation

The new equation then becomes \((x+8)^{2} = 81\)

Step 4: Solve for x

Once we have a perfect square on one side, we can solve for x by taking the square root of both sides: \(x + 8 = \pm \sqrt{81} \) which then simplifies to \(x + 8 = \pm 9\)

Step 5: Find the solutions

This gives us two possible solutions, \(x = 9 - 8 = 1\) or \(x = -9 -8 = -17\)

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra, typically expressed in the form \( ax^2 + bx + c = 0 \). These equations are called 'quadratic' because the highest power of the variable \(x\) is 2. Quadratic equations often appear in various scientific and engineering applications, such as physics, economics, and biology.

Understanding quadratic equations is essential because they can be applied to model real-world situations. For instance, predicting the trajectory of a projectile or calculating areas are based on quadratic principles. The main components of these equations are:

• \(a\): the coefficient of \(x^2\), which affects the width and direction of the parabola graph.
• \(b\): the coefficient of \(x\), which influences the position of the vertex.
• \(c\): the constant term, affecting the parabola's vertical shift.

Recognizing these parts helps in solving and rearranging quadratic equations effectively.

Perfect Squares

The concept of perfect squares is crucial when solving quadratic equations by completing the square. A perfect square trinomial looks like \( (x + a)^2 \), which expands to \( x^2 + 2ax + a^2 \). By transforming a quadratic equation into a perfect square trinomial, we can simplify the equation significantly.

To complete the square, we take the coefficient of \(x\) (represented by \(b\) in \( ax^2 + bx + c = 0 \)), divide it by 2, and then square the result. For instance, if \(b = 16\), then \( (16/2)^2 = 64 \). This process turns \( x^2 + 16x + 64 \) into \( (x + 8)^2 \).

Using perfect squares makes it possible to solve a quadratic equation that might initially seem daunting. This method works efficiently, providing a straightforward path from a convoluted quadratic expression to an easier form.

Solving Equations

Solving quadratic equations by completing the square is a method that allows us to find the values of \(x\) that satisfy the equation. To solve the equation \( x^2 + 16x = 17 \) by completing the square, we first transform it into a perfect square form.

After adding and rearranging terms in the previous steps to make a perfect square, the equation becomes \( (x + 8)^2 = 81 \). We can now solve this equation by taking the square root of both sides. This gives us two possible equations: \( x + 8 = 9 \) and \( x + 8 = -9 \).

Solving each results in the solutions \( x = 1 \) and \( x = -17 \), indicating that there are two points where the original quadratic equation equals zero. Understanding this process of solving equations helps build a solid foundation for tackling more complicated mathematical problems.