However, the equation x^3 + 8 = 0, for example, can be solved using factoring and the quadratic formula. In Chapter 6 we will discuss such higher degree equations in more detail. The equation in Example 7 is called a cubic equation, because of the term of degree 3. SQUARE ROOT PROPERTY The solution set of x^2=k is. The solution set is (1/3, -3/2).Ī quadratic equation of the form x^2 = k can be solved by factoring with the following sequence of equivalent equations. Check these solutions by substituting in the original equation. Solve each of these linear equations separately to find that the solutions of the original equation are 1/3 and -3/2. The next example shows how the zero-factor property is used to solve a quadratic equation.įirst write the equation in standard form asīy the zero-factor property, the product (3r -1)(2r + 3) can equal 0 only if If a and b are complex numbers, with ab = 0, then a = 0 or b=0 or both This method depends on the following property. The simplest method of solving a quadratic equation, but one that is not always easily applied, is by factoring. (Why is the restriction a!=0 necessary?) A quadratic equation written in the form ax^2+bx+c=0 is in standard form. Where a,b, and c are real numbers with a!=0, is a quadratic equation. A quadratic equation is defined as follows.Īn equation that can be written in the form Together you can come up with a plan to get you the help you need.As mentioned earlier, an equation of the form ax + b = 0 is a linear equation. See your instructor as soon as you can to discuss your situation. You should get help right away or you will quickly be overwhelmed. …no - I don’t get it! This is a warning sign and you must not ignore it. Is there a place on campus where math tutors are available? Can your study skills be improved? Whom can you ask for help?Your fellow classmates and instructor are good resources.
It is important to make sure you have a strong foundation before you move on. In math every topic builds upon previous work. This must be addressed quickly because topics you do not master become potholes in your road to success. What did you do to become confident of your ability to do these things? Be specific. Reflect on the study skills you used so that you can continue to use them. Congratulations! You have achieved the objectives in this section. Ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.Ĭhoose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.” Since these equations are all of the form x 2 = k, the square root definition tells us the solutions are the two square roots of k. If n 2 = m, then n is a square root of m. We earlier defined the square root of a number in this way:
So, every positive number has two square roots-one positive and one negative. Therefore, both 13 and −13 are square roots of 169. Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. īut what happens when we have an equation like x 2 = 7? Since 7 is not a perfect square, we cannot solve the equation by factoring. In each case, we would get two solutions, x = 4, x = −4 x = 4, x = −4 and x = 5, x = −5. We can easily use factoring to find the solutions of similar equations, like x 2 = 16 and x 2 = 25, because 16 and 25 are perfect squares. Let’s review how we used factoring to solve the quadratic equation x 2 = 9. We have already solved some quadratic equations by factoring. Solve Quadratic Equations of the form a x 2 = k a x 2 = k using the Square Root Property