{"id":2439,"date":"2024-10-09T22:16:42","date_gmt":"2024-10-09T22:16:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/?post_type=chapter&p=2439"},"modified":"2024-10-09T22:18:25","modified_gmt":"2024-10-09T22:18:25","slug":"other-removed-portions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/other-removed-portions\/","title":{"raw":"Other Removed Portions","rendered":"Other Removed Portions"},"content":{"raw":"

REMOVED FROM SECTION 3-1 (since it's in 2-2)<\/h2>\r\n

The Discriminant<\/h2>\r\nThe quadratic formula<\/strong> not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions. When we consider the discriminant<\/strong>, or the expression under the radical, [latex]{b}^{2}-4ac[\/latex], it tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. In turn, we can then determine whether a quadratic function has real or complex roots. The table below\u00a0relates the value of the discriminant to the solutions of a quadratic equation.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Value of Discriminant<\/th>\r\nResults<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]{b}^{2}-4ac=0[\/latex]<\/td>\r\nOne repeated rational solution<\/td>\r\n<\/tr>\r\n
[latex]{b}^{2}-4ac>0[\/latex], perfect square<\/td>\r\nTwo rational solutions<\/td>\r\n<\/tr>\r\n
[latex]{b}^{2}-4ac>0[\/latex], not a perfect square<\/td>\r\nTwo irrational solutions<\/td>\r\n<\/tr>\r\n
[latex]{b}^{2}-4ac<0[\/latex]<\/td>\r\nTwo complex solutions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n

A General Note: The Discriminant<\/h3>\r\nFor [latex]a{x}^{2}+bx+c=0[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers, the discriminant<\/strong> is the expression under the radical in the quadratic formula: [latex]{b}^{2}-4ac[\/latex]. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.\r\n\r\n<\/div>\r\n
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Example<\/h3>\r\nUse the discriminant to find the nature of the solutions to the following quadratic equations:\r\n
    \r\n \t
  1. [latex]{x}^{2}+4x+4=0[\/latex]<\/li>\r\n \t
  2. [latex]8{x}^{2}+14x+3=0[\/latex]<\/li>\r\n \t
  3. [latex]3{x}^{2}-5x - 2=0[\/latex]<\/li>\r\n \t
  4. [latex]3{x}^{2}-10x+15=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"497176\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497176\"]\r\n\r\nCalculate the discriminant [latex]{b}^{2}-4ac[\/latex] for each equation and state the expected type of solutions.\r\n
      \r\n \t
    1. [latex]\\begin{align}{x}^{2}+4x+4=0&&{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0\\end{align}\\hspace{4mm}[\/latex]There will be one repeated rational solution.<\/li>\r\n \t
    2. [latex]\\begin{align}8{x}^{2}+14x+3=0&&{b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100\\end{align}\\hspace{4mm}[\/latex]As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t
    3. [latex]\\begin{align}3{x}^{2}-5x - 2=0&&{b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49\\end{align}\\hspace{4mm}[\/latex]As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t
    4. [latex]\\begin{align}3{x}^{2}-10x+15=0&&{b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80\\end{align}\\hspace{4mm}[\/latex]There will be two complex solutions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe have seen\u00a0that a quadratic equation may have two real solutions, one real solution, or two complex solutions.\r\n\r\nLet\u2019s summarize\u00a0how the discriminant affects the evaluation of [latex] \\sqrt{{{b}^{2}}-4ac}[\/latex], and how it helps to determine the solution set.\r\n
        \r\n \t
      • If [latex]b^{2}-4ac>0[\/latex], then the number underneath the radical will be a positive value. You can always find the square root of a positive, so evaluating the Quadratic Formula will result in two real solutions (one by adding the square root, and one by subtracting it).<\/li>\r\n \t
      • If [latex]b^{2}-4ac=0[\/latex], then you will be taking the square root of 0, which is 0. Since adding and subtracting 0 both give the same result, the \"[latex]\\pm[\/latex]\" portion of the formula doesn't matter. There will be one real repeated solution.<\/li>\r\n \t
      • If [latex]b^{2}-4ac<0[\/latex], then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.<\/li>\r\n<\/ul>\r\n
        \r\n

        Example<\/h3>\r\nUse the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has.\r\n\r\n[reveal-answer q=\"116245\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"116245\"]\r\n\r\nEvaluate [latex]b^{2}-4ac[\/latex]. First note that\u00a0[latex]a=1,b=\u22124[\/latex], and [latex]c=10[\/latex].\r\n

        [latex]{c}b^{2}-4ac[\/latex]<\/p>\r\n

        [latex]\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)[\/latex]<\/p>\r\n

        [latex]16\u201340=\u221224[\/latex]<\/p>\r\nThe result is a negative number. The discriminant is negative, so [latex]x^{2}-4x+10=0[\/latex] has two complex solutions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

        \r\n

        Try It<\/h3>\r\n