Problem 69 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square or by using the quadratic formula. $$ r^{2}+2 r=9 $$

Step-by-Step Solution

Answer

Answer: The two solutions for the equation are $$r_1 = -1 + \sqrt{10}$$ and $$r_2 = -1 - \sqrt{10}$$.

Step 1: Set up the equation for completing the square

We want the right side of the equation to be 0, so let's move the constant over: $$ r^{2} + 2r - 9 = 0 $$ The equation is in the form $$ar^2 + br + c = 0$$ with $$a=1, b=2,$$ and $$c=-9$$.

Step 2: Completing the square

To complete the square, we need to add and subtract a value that makes the left side a perfect square trinomial. The value we should add and subtract is $$\left(\frac{b}{2a}\right)^2$$. For this equation, we have: $$ \left(\frac{2}{2 \cdot 1}\right)^2 = (1)^2 = 1 $$ Now let's add and subtract this value within the equation: $$ (r^2 + 2r + 1) - 1 - 9 = 0 $$ Simplify: $$ (r^2 + 2r + 1) - 10 = 0 $$ The left side of the equation will become a perfect square trinomial: $$ (r+1)^2 - 10 = 0 $$

Step 3: Solving the equation

Now, let's solve for $$r$$: $$(r+1)^2 - 10 = 0$$ Add 10 to both sides: $$(r+1)^2 = 10$$ Take the square root of both sides: $$\pm\sqrt{(r+1)^2} = \sqrt{10}$$ $$r + 1 = \pm\sqrt{10}$$ Now we can find the two values of r: $$r_1 = -1 + \sqrt{10}$$ $$r_2 = -1 - \sqrt{10}$$ So, the two solutions for the equation $$r^2+2r=9$$ are: $$r_1 = -1 + \sqrt{10}$$ $$r_2 = -1 - \sqrt{10}$$

Key Concepts

Completing the Square

Completing the square is a valuable technique for solving quadratic equations that makes the equation into a perfect square trinomial, which simplifies the process of finding the solution. To complete the square, you want to transform an equation of the form \( ax^2 + bx + c = 0 \) into something that looks like \((x + d)^2 = e\). Here's the basic idea:

• First, try to rearrange the equation so that the quadratic and linear terms are on one side, and the constant is on the other. For example, \( r^2 + 2r - 9 = 0 \) becomes \( r^2 + 2r = 9 \).
• Next, take the coefficient of the linear term (\( b \)), divide it by 2, and then square it. This value will help form a perfect square trinomial. In our case, \( b = 2 \), so you get \( \left(\frac{2}{2}\right)^2 = 1 \).
• Add and subtract this squared value on the side with the quadratic and linear terms: \( r^2 + 2r + 1 - 1 = 9 \).
• Simplify it to form a complete square: \( (r + 1)^2 \).
• The equation now is \( (r + 1)^2 - 10 = 0 \).

By completing the square, we have transformed a complicated quadratic equation into one that's much easier to solve.

Quadratic Formula

The quadratic formula is another very useful method for solving quadratic equations. It provides a straightforward way to find the solutions of any quadratic equation and can be applied when other methods are difficult to use. In standard form, a quadratic equation looks like \( ax^2 + bx + c = 0 \). The quadratic formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

When using the quadratic formula, just plug in the values of \( a \), \( b \), and \( c \) from your equation. For the given equation \( r^2 + 2r - 9 = 0 \):

• Here \( a = 1 \), \( b = 2 \), and \( c = -9 \).
• Substitute these values into the formula:\( r = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} \), which simplifies to \( r = \frac{-2 \pm \sqrt{40}}{2} \).
• Simplify further to find \( r = -1 \pm \sqrt{10} \).

This shows that the solutions to the equation are \( r_1 = -1 + \sqrt{10} \) and \( r_2 = -1 - \sqrt{10} \). This method is especially helpful for equations that are not easily factorable.

Perfect Square Trinomial

A perfect square trinomial results when an expression of the form \( (x + y)^2 \) is expanded. Completing the square helps to reshape a quadratic equation into a format that makes it recognizable as a perfect square trinomial. The benefits are:

• Recognition: A perfect square trinomial looks like \( a^2 + 2ab + b^2 \), allowing you to easily rewrite it as \( (a + b)^2 \).
• Simplification: With a perfect square trinomial, you can solve quadratic equations by simply "taking the square root" of the trinomial, significantly simplifying the problem.

In the given problem, when we completed the square, the expression \( r^2 + 2r + 1 \) was rewritten as \( (r + 1)^2 \). This trinomial is "perfect" because it directly factors back into \( (r + 1)^2 \). Recognizing a pattern helps students to approach solving quadratic equations more intuitively and reduces the chances of mistakes.