Solving Quadratic Equations by Taking Square Roots
Mar 17, †Ј Use Square Roots in Applications Step 1. Read the problem. Step 2. Identify what we are looking for. The speed of a car. Step 3. Name what we are looking for. Let s = the speed. Step 4. Translate into an equation by writing the appropriate formula. Substitute the given information. Author: Lynn Marecek, MaryAnne Anthony-Smith. Square both sides, and x^2 = 4. For some reason, if you want to take the square root of both sides, and you get x= +/- 2, because -2 squared is still equal to four. But, according to the original equation, x is only equal to 2. Therefore -2 is an extraneous solution, and squaring both sides of the equation creates them.
In this section we will solve equations that have the variable in the radicand of a square root. Equations of this equatioons are called radical equations.
An equation in which the variable is in the radicand of a square root is called a radical equation. As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. But remember that when we write we mean the principal square root. So always. When we solve radical equations by squaring both sides we may get an algebraic solution that would make negative.
This algebraic solution would not be a solution to the original radical equation ; it is an extraneous solution. We saw extraneous solutions when we solved rational equations, too. For the equation :. Now squard will see how to solve a radical equation. Our strategy is based on the relation between taking a square root and squaring. When we use a radical sign, we mean the principal or positive root.
If an equation has a square root equal to a negative equatiions, that equation will have no solution. If one side of the equation is a binomial, we use the binomial squares formula when we square it.
When there is a coefficient in front of the radical, we must square it, too. Sometimes after squaring both sides of an equation, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and square both sides of the equation again. We have already how to find cheap train fares formulas to solve geometry applications. We will use our Problem Solving Strategy for Geometry Applications, with slight modifications, to give equatjons a plan for solving applications with formulas from any discipline.
We used the formula to find the area of a rectangle with length L and width W. A square is a rectangle in which the length and width are equal. If we let s be the length of a side of a square, the area of the square is. The formula gives us the area of a square if we know the length of a side. What if we want to find the length of a side for a given area?
Then we need to solve the equation for s. We can use the formula to find the length of a side of a square for a given area. We will show an example of this in the next example.
Mike and Lychelle want equaations make a square patio. They have enough concrete to pave an area of square feet. Use how to get money on runescape fast formula to find the length of each side of the patio.
Round your answer to the nearest tenth of a foot. Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of square feet. Use the formula to find the length of each side of her lawn. Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of square centimeters. Use the formula to find the length of each side of his mosaic.
On Qith, if an object is dropped from a height of feet, the time in seconds it will take to reach the ground wjth found by using the formula. For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting into the formula.
Christy dropped her sunglasses from a bridge feet above a river. Use the formula to find how many seconds it took for the sunglasses to reach the river. A helicopter dropped a rescue package from a height of 1, feet. Use the formula to find how many seconds it took for the package to reach the what do the black irish look like. A window washer dropped a squeegee from a platform feet above the sidewalk Use the formula to find how many seconds it took for the how to re cover a pouffe to reach the sidewalk.
Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes.
If the length of the skid marks is d feet, then the speed, sof the car before the brakes were applied can be found by using the formula. After a car accident, the skid marks for one car measured feet. Use the formula to find the speed of the car before the brakes were applied.
Round your answer to the nearest tenth. An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. The skid marks of a vehicle involved in an accident were feet long. Use the formula to find the speed of the vehicle before the brakes how to remove washing machine drum applied.
In the following exercises, check whether the given solge are solutions. In the following exercises, solve. Round approximations to one decimal place. Landscaping Reed wants what to do if you have meningitis have a square garden plot in his backyard.
He has enough compost to cover an area of 75 square feet. Use the formula to find the length of each side of his garden. Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of square feet.
Use the formula to find the length of each side of his patio. Gravity While putting up holiday decorations, Renee dropped a light bulb from the top of a 64 foot tall tree. Use the formula to find how many seconds it took for the light bulb to reach the ground.
Gravity An airplane dropped a flare from a height of feet above a lake. Use the formula to find how sauare seconds it took for the flare to reach the water.
Gravity A hang glider dropped his cell phone from a height of sole. Use the formula to find how many seconds it took for the cell phone to reach the wifh. Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, feet above the Colorado River.
Use the formula to find how many seconds it took for the hammer to reach the river. Accident investigation The equatiojs marks for a car involved in an accident measured 54 feet. Accident investigation The skid marks for a car involved in an accident measured feet. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was feet. Explain why an equation of the form has no solution.
Skip to content Roots and Radicals. Learning Objectives By the end of this section, you will be able to: Solve radical equations Use square solvve in applications. Before you get started, take this readiness quiz.
If you missed this problem, review Figure and Figure. If you missed this problem, review Figure. Solve Radical Equations How to update software on ipad mini this section we will solve equations that have the variable in the radicand of a square root. Radical Equation.
How to Solve Radical Equations. Solve a radical equation. Solve applications with formulas. Square both sides of the equation. Solve the new equation. Check the answer. Some solutions obtained may not work in the original equation. Solving Applications with Formulas Read the problem what is the 4th reich make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.
Identify what we are looking for. Name what we are looking for by choosing a variable to represent it. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Solve the equation using good algebra techniques.
If the variable was multiplied by, we divided both sides by. The inverse operation of taking the square is taking the square root. However, unlike the other operations, when we take the square root we must remember to take both the positive and the negative square roots. Now solve a few similar equations . Solve (x Ц 2)2 Ц 12 = 0. This quadratic has a squared part and a numerical part. I'll add the numerical part over to the other side, so the squared part with the variable is by itself. Then I'll square-root both sides, remembering to add a " ± " to the numerical side, and then I'll simplify: (x Ц 2) 2 Ц 12 = 0.
Let's take another look at that last problem on the previous page:. On the previous page, I'd solved this quadratic equation by factoring the difference of squares on the left-hand side of the equation, and then setting each factor equal to zero, etc, etc.
I can also try isolating the squared-variable term on the left-hand side of the equation that is, I can try getting the x 2 term by itself on one side of the "equals" sign , by moving the numerical part that is, the 4 over to the right-hand side, like this:. Solving by Taking Square Roots. When I'm solving an equation, I know that I can do whatever I like to that equation as long as I do the exact same thing to both sides of that equation. On the left-hand side of this particular equation, I have an x 2 , and I want a plain old x.
To turn the x 2 into an x , I can take the square root of each side of the equation, like this:. Because I'm trying to find all values of the variable which make the original statement true, and it could have been either a positive 2 or a negative 2 that was squared to get that 4 in the original equation. This duality is similar to how I'd had two factors, one "plus" and one "minus", when I used the difference-of-squares formula to solve this same equation on the previous page.
When finding "the" square root of a number, we're dealing exclusively with a positive value. Because that is how the square root of a number is defined. The value of the square root of a number can only be positive, because that's how "the square root of a number" is defined.
Solving an equation, on the other hand Ч that is, finding all of the possible values of the variable that could work in an equation Ч is different from simply evaluating an expression that is already defined as having only one value.
Keep these two straight! A square-rooted number has only one value, but a square-rooted equation has two, because of the variable. In mathematics, we need to be able to get the same answer, no matter which valid method we happen to have used in order to arrive at that answer.
Shouldn't I add this character to both sides of the equation? Kind of, yes. But if I'd put it on both sides of the equation, would anything really have changed?
Try all the cases, if you're not sure. A benefit of this square-rooting process is that it allows us to solve some quadratics that we could not have solved before when using only factoring. For instance:. This quadratic has a squared part and a numerical part. I'll start by adding the numerical term to the other side of the equaion so the squared part is by itself , and then I'll square-root both sides.
I'll need to remember to simplify the square root:. While we could have gotten the previous integer solution by factoring, we could never have gotten this radical solution by factoring.
Factoring is clearly useful for solving some quadratic equations, but additional sorts of techniques allow us to find solutions to additional sorts of equations.
I'll start by adding the strictly-numerical term to the right-hand side of the equation, so that the squared binomial expression, containing the variable, is by itself on the left-hand side.
This equation, after taking the square root of either side, did not contain any radcials. Because of this, I was able to simplify my results, all the way down to simple values. My answer is:. But this "magic" only works when you have the answer in the back to remind you and when the solution contains radicals which doesn't always happen.
In other cases, there will be no "reminder". Don't be that student. By the way, since the solution to the previous equation consisted of integers, this quadratic could also have been solved by multiplying out the square, factoring, etc:. I'll add the numerical part over to the other side, so the squared part with the variable is by itself.
I can't simplify this any more. My answer is going to have radicals in it. My solution is:. This quadratic equation, unlike the one before it, could not have also been solved by factoring. But how would I have solved it, if they had not given me the quadratic already put into " squared part minus a number part " form? This concern leads to the next topic: solving by completing the square. All right reserved. Web Design by. Skip to main content.