Extract Square Roots

Extracting Square Roots Recall that a quadratic equation is in standard form!!if it is equal to 0: where a, b, and c are real numbers and . A solution to such an equation is called a root. Quadratic equations can have two real solutions, one real solution, or no real solution. If the quadratic expression on the left factors, then we can solve it by factoring. A review of the steps used to solve by factoring follow: Step 1: Express the quadratic equation in standard form. Step 2: Factor the quadratic expression. Step 3: Apply the zero-product property and set each variable factor equal to 0. Step 4: Solve the resulting linear equations. For example, we can solve by factoring as follows: The two solutions are -2 and 2. The goal in this section is to develop an alternative method that can be used to easily solve equations where , giving the form The equation is in this form and can be solved by Þrst isolating If we take the square root of both sides of this equation, we obtain the following: Here we see that x=-2 and x=2 are solutions to the resulting equation. In general, this describes the!square root property; for any real number k, The notation Ò±Ó is read Òplus or minusÓ and is used as compact notation that indicates two solutions. Hence the statement indicates that or . Applying the square root property as a means of solving a quadratic equation is called extracting the roots. Example 3: Solve: . Solution: Begin by isolating Examples Example 1: Solve: Solution: Begin by isolating the square. Next, apply the square root property. Answer: The solutions are -5 and 5. The check is left to the reader. Certainly, the previous example could have been solved just as easily by factoring. However, it demonstrates a technique that can be used to solve equations in this form that do not factor. Example 2: Solve: Solution: Notice that the quadratic expression on the left does not factor. We can extract the roots if we Þrst isolate the leading term, Apply the square root property. For completeness, check that these two real solutions solve the original quadratic equation. Generally, the check is optional. Answer: The solutions are Example 3: Solve: Solution: Begin by isolating and Apply the square root property and then simplify. Answer: The solutions are and Example 4: Solve: Solution: Begin by isolating After applying the square root property, we are left with the square root of a negative number. Therefore, there is no real solution to this equation. Answer: No real solution Reverse this process to Þnd equations with given solutions of the form ± k. Example 5: Find an equation with solutions and Solution: Begin by squaring both sides of the following equation: Lastly, subtract 12 from both sides and present the equation in standard form. Answer: Solve by Extracting the Roots Consider solving the following equation: To solve this equation by factoring, Þrst square x + 2 and then put it in standard form, equal to zero, by subtracting 25 from both sides. Factor and then apply the zero-product property. The two solutions are -7 and 3. When an equation is in this form, we can obtain the solutions in fewer steps by extracting the roots. Examples Example 1: Solve: Solution: Solve by extracting the roots. At this point, separate the Òplus or minusÓ into two equations and simplify each individually. Answer: The solutions are -7 and 3. In addition to fewer steps, this method allows us to solve equations that do not factor. Example 2: Solve: Solution: Begin by isolating the square. Next, extract the roots and simplify. Solve for x. Answer: The solutions are and Example 3: Solve: Solution: Begin by isolating the square factor. Apply the square root property and solve. Answer: The solutions are and Geometry Example Example 1: The length of a rectangle is twice its width. If the diagonal measures 2 feet, then Þnd the dimensions of the rectangle. Solution: The diagonal of any rectangle forms two right triangles. Thus the Pythagorean theorem applies. The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse: Solve. Here we obtain two solutions, and . Since the problem asked for a length of a rectangle, we disregard the negative answer. Furthermore, we will rationalize the denominator and present our solutions without any radicals in the denominator. Back substitute to Þnd the length. Answer: The length of the rectangle is feet and the width is Questions Question 1 Solve Solution The answer is!n=7 and n=-7 Begin by applying the square root property. Question 2 Solve! Solution The answer is!! and! Begin by applying the square root property. Question 3 Solve Solution The answer is!! and! Begin by isolating the square. Next, apply the square root property. This gives us two answers: and Question 4 Solve! Solution The answer is!! and! Begin by isolating the square. Next, apply the square root property. Question 5 Solve Solution The answer is!! and! Begin by isolating the square. Next, apply the square root property. Question 6 Solve! Solution The answer is!! and! Begin by isolating the square. Next, apply the square root property. This gives us two answers: and Question 7 Solve by factoring! Solution This gives us two answers: and Summary! by extracting the roots. ¥ Solve equations of the for m ¥ Extracting roots involves isolating the square and then applying the square root property. After applying the square root property, you have two linear equations that each can be solved. Be sure to simplify all radical expressions and rationalize the denominator if necessary.