Questionsįactor each of the following polynomials and solve what you can. Checking for any others by using the discriminant reveals that all other solutions are complex or imaginary solutions. The two real solutions are x = 2 and x = -1. The factored (x^3 - 8) and (x^3 + 1) terms can be recognized as the difference of cubes. Now that the substituted values are factored out, replace the u with the original x^3. Here, it would be a lot easier if the expression for factoring was x^2 - 7x - 8 = 0.įirst, let u = x^3, which leaves the factor of u^2 - 7u - 8 = 0. This same strategy can be followed to solve similar large-powered trinomials and binomials.įactor the binomial x^6 - 7x^3 - 8 = 0. Solving each of these terms yields the solutions x = \pm 3, \pm 2. This is done using the difference of squares equation: a^2 - b^2 = (a + b)(a - b).įactoring (x^2 - 9)(x^2 - 4) = 0 thus leaves (x - 3)(x + 3)(x - 2)(x + 2) = 0. To complete the factorization and find the solutions for x, then (x^2 - 9)(x^2 - 4) = 0 must be factored once more. Once the equation is factored, replace the substitutions with the original variables, which means that, since u = x^2, then (u - 9)(u - 4) = 0 becomes (x^2 - 9)(x^2 - 4) = 0. Now substitute u for every x^2, the equation is transformed into u^2-13u+36=0. The area of a rectangular garden is 30 square feet. For example, 12x2 + 11x + 2 7 must first be changed to 12x2 + 11x + 5 0 by subtracting 7 from both sides. There is a standard strategy to achieve this through substitution.įirst, let u = x^2. When you use the Principle of Zero Products to solve a quadratic equation, you need to make sure that the equation is equal to zero. Here, it would be a lot easier when factoring x^2 - 13x + 36 = 0. Since \(D=0\), this tells us that \(x^2-2x+1=0\) only has one root.Solve for x in x^4 - 13x^2 + 36 = 0.įirst start by converting this trinomial into a form that is more common. To calculate the discriminant, we plug in \(a=1, b=-2, c=1\) into the discriminant formula: Let's apply this idea to our previous example: \(x^2-2x+1=0\). Tip: Make sure that the quadratic equation you are working with is written in \(ax^2+bx+c=0\) form before calculating its discriminant! To determine the number of roots a quadratic equation has, we can use a part of the quadratic formula called the discriminant: this quadratic equation only has one root). In fact, it is the only root of this equation (i.e. In our previous examples, you might have noticed that some equations had a different number of roots/solutions - 0 roots, 1 root or 2 roots.įor example, for \(x^2-2x+1=0\), we mentioned that \(x=1\) is a root/solution to this quadratic equation. So, we actually have two pairs of numbers that work in the given statement: Since \(x+y=176\), we can rearrange this equation and use it to find \(y\):Ĭhecking our work that \(y=x^2\), indeed \(163.22 \approx (12.78)^2\) Now that we our solutions, we can plug them back into the original equations to find the values for \(y\), as well as check our work to make sure our solutions are valid. Since this equation does not easily factor, we apply the Quadratic Formula to find the solutions: To determine its solutions, we need to make one side equal to 0, then factor it: Notice that we now have a quadratic equation. Now, we can substitute the first equation into the second to end up with one equation we will solve: Since "their sum is 176", we have the equation: Since we're given that "one number is the square of another", if we let \(x\) represent one number, and \(y\) represent the other number, we have the equation representing their relationship: If their sum is 176, what are the two numbers? Round answers to two decimal places. Infusion Rates for Intravenous Piggyback (IVPB) BagĮxample: One number is the square of another number.
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