All Algebra II Resources
Example Questions
Example Question #24 : Completing The Square
Solve the equation by completing the square:
In order to complete the square, we must get all the terms with x in them to one side of the equation. For this problem, we subtract 9 on both sides in order to yield:
.
Remember that completing the square means that we take half of the coefficient of x (), square this new value (), and add it to both sides of the equation. Our equation now looks like:
The left side is a perfect square polynomial, as we have set it up this way. We factor it as such:
After some cleanup, we arrive at:
In order to solve for x, we must take the square root of both sides of the equation.
Finally, we add 6 to both sides of the equation, and simplify the square root of 27:
Example Question #93 : Solving Quadratic Equations
Put the quadratic equation into vertex form by completing the square.
To complete the square, first set our equation equal to 0:
add 7 to both sides
we want to find the number we can add to both sides so that the left side can be factored as one binomial squared. This binomial must be , since when multiplied by itself you'd end up adding which is what we need. Multiplying:
so the number we want to add to both sides is 6.25
we constructed the left side so we could re-write it as:
simplifying the right gives
now we can subtract 13.25 from both sides to get 0 again:
so our equation is
Example Question #94 : Solving Quadratic Equations
Put into vertex form by completing the square.
To complete the square, first set the equation equal to 0:
subtract 1 from both sides
factor out the 2 on the left side
Now we're trying to figure out what we can add in that space so that the expression in parentheses can be factored as a binomial squared. works as the binomial since we know we will be adding . Squaring this yields:
So the number we want to add is 1. BUT BE CAREFUL! We're adding a 1 to inside those parentheses, so really, we're adding 2, since we distribute that 2. Add 2 to both sides:
we can easily simplify the right side, plus we know that we can factor the left since we set it up to be able to:
now subtract 1 from both sides
so our equation is
Example Question #27 : Completing The Square
Find the roots of this quadratic equation by completing the square:
This equation does not have x intercept (s).
To solve a quadratic with an "a" term of 1 (from the standard form ) by completing the square you must first move over the constant. Next, halve the "b" term, square it, and add to both sides. Then factor the left side and set it equal to the constant. Note that the factor of the quadratic you "made" will always be of the format where the sign is the original sign of the b term.
Move the constant:
Halve the b term, square it, and add to both sides:
Factor the left side and simplify the right:
Take the square root of both sides:
Solve for x:
Example Question #28 : Completing The Square
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?
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.
Example Question #29 : Completing The Square
Re-write this quadratic in vertex form by completing the square:
First, factor out the 2 from the first 2 terms:
add 3 to both sides
inside the parentheses, add
since the 4 was added in the parentheses, it's multiplied by 2. That means we added 8, so add 8 to the other side too
simplify by re-writing the left and adding 3 and 8 on the right
subtract 11 from both sides
Example Question #101 : Solving Quadratic Equations
Ahmed is trying to solve the equation by completing the square. His first two steps are shown below:
Step 1:
Step 2:
Ahmed knows that he needs to add a number to both sides in the next step.
What number should Ahmed add to both sides?
A perfect square has the form .
In this case, , so is or .
To square a fraction, simply square the numerator and the denominator: .
Example Question #421 : Intermediate Single Variable Algebra
Solve the equation by completing the square.
or
or
To solve the equation by completing the square first move the constant term to the right hand side of the equation.
Now, remember to divide the middle term by two. Then square it and add it to both sides of the equation.
From here write the the middle term divided by two in a binomial expression
Square root both sides and recall that .
Example Question #103 : Solving Quadratic Equations
Solve .
Solve by completing the square
add to both sides, where .
Factor
Example Question #421 : Intermediate Single Variable Algebra
Solve by completing the square:
Add 7 to both sides:
Divide both sides by the coefficient on x^2:
Add to both sides:
Form the perfect square on the left side:
Simplify the right side:
Take the square root of both sides:
Solve for x: