Chapter One
Section 9.2Solving Quadratic Equations by the Quadratic Formula Objective 1: Solving Quadratic Equations by Using the Quadratic FormulaAny quadratic equation can be solved by completing the square. By completing the square for a general quadratic equation ax2+bx+c=0, we derive the quadratic formula. ax2+bx+c=0x2+bax+ca=0x2+bax=-cax2+bax+b24a2=-ca+b24a2x+b2a2=b2-4ac4a2x+b2a=±b2-4ac4a2x=-b2a±b2-4ac2ax=-b±b2-4ac2aBecause the quadratic formula is derived by completing the square, it can be used to solve any quadratic equation written in standard form. Quadratic Formula:A quadratic equation written in the form ax2+bx+c=0 has the solutions -b±b2-4ac2a.Use the quadratic formula to solve the equation. Give the answers in exact form using simplified radicals and i as needed.a. x2+8x+25=0b. 2x2=3x+4Objective 2: Using the DiscriminantIn the quadratic formula, the value of the radicand, b2-4ac, tells us the number and type of solutions of the corresponding quadratic equation. This value is called the discriminant. Discriminant:For a quadratic equation of the form ax2+bx+c=0, when b2-4ac>0, the quadratic equation has two real solutions.when b2-4ac=0, the quadratic equation has one real solution.when b2-4ac<0, the quadratic equation has two complex, nonreal solutions.a. Determine the number and types of solutions of the quadratic equation 5=4x-3x2.Recall that the solution(s) of the quadratic equation ax2+bx+c=0 correspond to the x-intercept(s) of the graph of the quadratic function fx=ax2+bx+c when the solutions are real numbers. Graph of fx=ax2+bx+c:When b2-4ac>0, the graph of f has two x-intercepts.When b2-4ac=0, the graph of f has one x-intercept.When b2-4ac<0, the graph of f has no x-intercepts.b. Determine the number of x-intercepts of the graph of fx=-2x2+12x-7.Objective 3: Solving Problems Modeled by Quadratic Equations The quadratic formula can be useful in solving problems that are modeled by quadratic equations.a. The base of a triangle is eight more than twice its height. If the area of the triangle is 55 square centimeters, find its base and height. b. A ball is thrown downward from the top of a 140-foot building with an initial velocity of 17 feet per second. The height of the ball h after t seconds is given by the equation h=-16t2-17t+140. How long after the ball is thrown will it strike the ground? ................
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