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Quadratic Equations Quadratic Equations and Problem Solvingand Problem Solving
♦ Understand basic concepts about quadratic Understand basic concepts about quadratic equationsequations
♦ Use factoring, the square root property, Use factoring, the square root property, completing the square, and the quadratic completing the square, and the quadratic formula to solve quadratic equationsformula to solve quadratic equations
♦ Understand the discriminantUnderstand the discriminant♦ Solve problems involving quadratic equationsSolve problems involving quadratic equations
Solving Quadratic EquationsQuadratic equations can have no real solutions, one real solution, or two real solutions.The are four basic symbolic strategies in which quadratic equations can be solved.
• Factoring• Square root property• Completing the square• Quadratic formula
FactoringFactoring is a common technique used to solve equations. It is based on the zero-product property, which states that if
ab = 0, then a = 0 or b = 0 or both. It is important to remember that this property works only for 0. For example, if ab = 1, then this equation does not imply that either a = 1 or b = 1. For example, a = 1/2 and b = 2 also satisfies ab = 1 and neither a nor b is 1.
If a metal ball is dropped 100 feet from a water tower, its height h in feet above the ground after t seconds is given by h(t) = 100 – 16t2. Determine how long it takes the ball to hit the ground.
Solution
The ball strikes the ground when the equation 100 – 16t2 = 0 is satisfied.
Another technique that can be used to solve a quadratic equation is completing the square. If a quadratic equation is written in the form x2 + kx =d, where k and d are constants, then the equation can be solved using
Quadratic equations can be solved symbolically, numerically, and graphically. The following example illustrates each technique for the equation x(x – 2) = 3.
If the quadratic equation ax2 + bx + c = 0 is solved graphically, the parabolay = ax2 + bx + c can intersect the x-axis zero, one, or two times. Each x-intercept is a real solution to the quadratic equation.
Use the discriminant to find the number of solutions to 9x2 – 12.6x + 4.41 = 0. Then solve the equation by using the quadratic formula. Support your answer graphically.
Many types of applications involve quadratic equations. To solve these problems, we use the steps for “Solving Application Problems” from Section 2.3 on page 122.
A box is being constructed by cutting2-inch squares from the corners of a rectangular piece of cardboard that is 6 inches longer than it is wide. If the box is to have a volume of 224 cubic inches, find
The dimensions can not be negative, so the width is 12 inches and the length is 6 inches more, or 18 inches.
Step 4: After the 2-inch-square corners are cut out, the dimensions of the bottom of the box are 12 – 4 = 8 inches by 18 – 4 = 14 inches. The volume of the box is then 2•8•14 = 224 cubic inches, which checks.