![]() The Discriminant Example Use the discriminant to determine the number and type of solutions for the following equation. The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively. The discriminant will take on a value that is positive, 0, or negative. The Discriminant The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant. ![]() x2 + 6x + 30 = 0 a = 1, b = 6, c = 30 So there is no real solution. The Quadratic Formula Example Solve x(x + 6) = 30 by the quadratic formula. The Quadratic Formula Example x2 + 8x – 20 = 0 (multiply both sides by 8) a = 1, b = 8, c = 20 Solve x2 + x – = 0 by the quadratic formula. The Quadratic Formula Example Solve 11n2 – 9n = 1 by the quadratic formula. The Quadratic Formula A quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions. The formula is derived from completing the square of a general quadratic equation. The Quadratic Formula Another technique for solving quadratic equations is to use the quadratic formula. Solving Quadratic Equations by the Quadratic Formula § 16.3 X2 + 7x + = ½ + = (x + )2 = Solving Equations Example Solve by completing the square. (y + ½)2 = Solving Equations Example Solve by completing the square. Y + 3 = ± = ± 1 Solving Equations Example Solve by completing the square. Complete the square (half the coefficient of the x term squared, added to both sides of the equation).Isolate all variable terms on one side of the equation.If the coefficient of x2 is NOT 1, divide both sides of the equation by the coefficient.It involves creating a trinomial that is a perfect square, setting the factored trinomial equal to a constant, then using the square root property from the previous section.Ĭompleting the Square Solving a Quadratic Equation by Completing a Square So, to find the constant term of a perfect square trinomial, we need to take the square of half the coefficient of the x term in the trinomial (as long as the coefficient of the x2 term is 1, as in our previous examples).Īdd Completing the Square Example What constant term should be added to the following expressions to create a perfect square trinomial? Ĭompleting the Square Example We now look at a method for solving quadratics that involves a technique called completing the square. Also, the constant on the left is the square of the constant on the right. Solving Quadratic Equations by Completing the Square § 16.2Ĭompleting the Square In all four of the previous examples, the constant in the square on the right side, is half the coefficient of the x term on the left. Square Root Property 3x – 17 = Example Solve (3x – 17)2 = 28 Square Root Property Example Solve (x + 2)2 = 25 x = 2 ± 5 x = 2 + 5 or x = 2 – 5 x = 3 or x = 7 Square Root Property Example Solve x2 + 4 = 0 x2 = 4 There is no real solution because the square root of 4 is not a real number. Square Root Property Example Solve x2 = 49 Solve 2x2 = 4 x2 = 2 Solve (y – 3)2 = 4 y = 3 2 y = 1 or 5 Square Root Property If b is a real number and a2 = b, then This chapter will introduce additional methods for solving quadratic equations. ![]() Square Root Property We previously have used factoring to solve quadratic equations. ![]() Solving Quadratic Equations by the Square Root Property § 16.1 Chapter Sections 16.1 – Solving Quadratic Equations by the Square Root Property 16.2 – Solving Quadratic Equations by Completing the Square 16.3 – Solving Quadratic Equations by the Quadratic Formula 16.4 – Graphing Quadratic Equations in Two Variables 16.5 – Interval Notation, Finding Domains and Ranges from Graphs and Graphing Piecewise-Defined Functions
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