Algebra II - Completing the Square

Which of the following is the same after completing the square?

$(x+\frac{1}{6})^2 =\frac{73}{36}$

$(x-\frac{1}{6})^2 =\frac{73}{36}$

$(x+\frac{1}{6})^2 =\frac{1}{2}$

$(x+\frac{1}{3})^2 =\frac{7}{2}$

$(x-\frac{1}{3})^2 =\frac{7}{2}$

Explanation

Divide by three on both sides.

$\frac{3x^2+x-6 }{3}=\frac{0}{3}$

$x^2+\frac{1}{3}x-2 = 0$

Add two on both sides.

$x^2+\frac{1}{3}x-2 +2= 0+2$

$x^2+\frac{1}{3}x = 2$

To complete the square, we will need to divide the one-third coefficient by two, which is similar to multiplying by one half, square the quantity, and add the two values on both sides.

$x^2+\frac{1}{3}x+ (\frac{1}{3}\cdot \frac{1}{2})^2 = 2+ (\frac{1}{3}\cdot \frac{1}{2})^2$

Simplify both sides.

$x^2+\frac{1}{3}x+ \frac{1}{36} = 2+ \frac{1}{36}$

Factor the left side, and combine the terms on the right.

$(x+\frac{1}{6})^2 = \frac{72}{36}+ \frac{1}{36}$

The answer is: $(x+\frac{1}{6})^2 =\frac{73}{36}$

Solve by completing the square:

$\small x^2-8x+9=0$

$\small x=4\pm\sqrt7$

$\small x=9\pm2\sqrt2$

$\small x=-9\pm2\sqrt2$

$\small x=-4\pm\sqrt7$

Explanation

To complete the square, the equation must be in the form:

$\small \small ax^2+2ab+b^2=c$

$\small x^2-8x+9=0$

$\small a=1$

$\small 2ab=-8$

$\small b=-4$

$\small b^2=16$

$\small x^2-8x+9+7=0+7$

$\small x^2-8x+16=7$

$\small (x-4)^2=7$

$\small \small x-4=\pm\sqrt7$

$\small x=4\pm\sqrt7$

What is/are the solution(s) to the quadratic equation

Hint: Complete the square

Explanation

When using the complete the square method we will divide the coefficient by two and then square it. This will become our term which we will add to both sides.

In the form,

our , and we will complete the square to find the value.

Therefore we get:

$x^2 + 4x+\left(\frac{4}{2}\right)^2 = 5+\left(\frac{4}{2}\right)^2$

$x+2=\pm3$

Josephine wanted to solve the quadratic equation below by completing the square. Her first two steps are shown below:

Equation:

Step 1:

Step 2:

Which of the following equations would best represent the next step in solving the equation?

Explanation

To solve an equation by completing the square, you must factor the perfect square. The factored form of is . Once the left side of the equation is factored, you may take the square root of both sides.

Solve for by completing the square.

$x=0.24\text{ and }x=-8.24$

$x=0.39\text{ and }x=-4.39$

$x=1.24\text{ and }x=-10.24$

$x=0.97\text{ and }x=-8.97$

Explanation

Start by adding to both sides so that the terms with the are together on the left side of the equation.

Now, look at the coefficient of the -term. To complete the square, divide this coefficient by , then square the result. Add this term to both sides of the equation.

Rewrite the left side of the equation in the squared form.

Take the square root of both sides.

$x+4=\pm \sqrt{18}$

Now solve for .

$x=-4+\sqrt{18}=0.24$

$x=-4-\sqrt{18}=-8.24$

Round to two places after the decimal.

Use completing the square to solve the following equation, simplifying radicals completely:

$x = - 3 \pm 2\sqrt{5}$

$x = -3 \pm3\sqrt{2}$

$x = -3 \pm \sqrt{11}$

$x = 3 \pm 2\sqrt{5}$

$x = -1 \pm 2\sqrt{5}$

Explanation

From the original equation, we add 18 to both sides in order to set up our "completing the square."

To make completing the square sensible, we divide both sides by 2.

$\frac{22}{2} = \frac{2x^2 + 12x}{2}$

We now divide the x coefficient by 2, square the result, and add that to both sides.

$\frac{6}{2} = 3$

Since the right side is now a perfect square, we can rewrite it as a square binomial.

Take the square root of both sides, simplify the radical and solve for x.

$\sqrt{20}=\sqrt{(x+3)^2}$

$\pm2\sqrt{5} = x+ 3$

$-3 \pm 2\sqrt{5} = x$

Solve the following equation by completing the square. Use a calculator to determine the answer to the closest hundredth.

and

No solution

Explanation

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

$\frac{4}{2}=2$

Square that and add it to both sides:

Now, you can factor the quadratic:

Take the square root of both sides:

$x+2 = \pm \sqrt{7}$

Finish out the solution:

$x = \pm\sqrt{7}-2$

$\sqrt{7}-2 = 0.64575131106459$

$-\sqrt{7}-2=-4.64575131106459$

Which of the following equations is equivalent to after completing the square?

Explanation

In order to complete the square, divide the coefficient in the equation by two and square the quantity.

$(\frac{4}{2})^2 = (2)^2=4$

This value must be added to both sides of the equation.

Factorize the left side and simplify the right side.

The answer is:

Solve for by completing the square.

$x=5.65\text{ and }x=0.35$

$x=10.55\text{ and }x=2.45$

$x=-8.75\text{ and }x=-2.25$

$x=1.95\text{ and }x=5.05$

Explanation

Start by subtracting from both sides so that the terms with the are together on the left side of the equation.

Now, look at the coefficient of the -term. To complete the square, divide this coefficient by , then square the result. Add this term to both sides of the equation.