4/21/2024 0 Comments Ways to solve quadratic equations![]() ![]() ![]() We will not attempt to prove this theorem but note carefully what it states. In other words, if the product of two factors is zero, then at least one of the factors is zero. The method of solving by factoring is based on a simple theorem. This method cannot always be used, because not all polynomials are factorable, but it is used whenever factoring is possible. The simplest method of solving quadratics is by factoring. It is possible that the two solutions are equal.Ī quadratic equation will have two solutions because it is of degree two. This theorem is proved in most college algebra books.Īn important theorem, which cannot be proved at the level of this text, states "Every polynomial equation of degree n has exactly n roots." Using this fact tells us that quadratic equations will always have two solutions. The solution to an equation is sometimes referred to as the root of the equation. In other words, the standard form represents all quadratic equations. The standard form of a quadratic equation is ax 2 + bx + c = 0 when a ≠ 0 and a, b, and c are real numbers.Īll quadratic equations can be put in standard form, and any equation that can be put in standard form is a quadratic equation. Solve a quadratic equation by factoring.Ī quadratic equation is a polynomial equation that contains the second degree, but no higher degree, of the variable.Place a quadratic equation in standard form.Upon completing this section you should be able to: QUADRATICS SOLVED BY FACTORING OBJECTIVES You now have the necessary skills to solve equations of the second degree, which are known as quadratic equations. In previous chapters we have solved equations of the first degree. All skills learned lead eventually to the ability to solve equations and simplify the solutions. Solving equations is the central theme of algebra. Solve a Quadratic Equation by COMPLETING THE SQUARE. To determine when the height of the ball is 336 feet. The distance along the ground from the bottom of the pole to the end of the wire is 4 feet greater than the height where the wire is attached to the pole. How far up the pole does the guy wire reach?Įxample 4: You throw a ball straight up from a rooftop 384 feet high with an initial speed of 3 feet per second. The functionĭescribes the height of the ball above the ground, s (t), in feet, t seconds after you threw it. The ball misses the rooftop on its way down and eventually strikes the ground. How long will it take for the ball to hit the ground? Eliminate any unreasonable answers.Įxample 2: Each side of a square is lengthened by 7 inches. The area of this new larger square is 81 square inches. Find the length of a side of the original square.Įxample 3: A guy wire is attached to a tree to help it grow straight. The length of the wire is 2 feet greater than the distance from the base of the tree to the stake. The height of the wooden part of the tree is 1 foot greater than the distance from the base of the tree to the stake.Įxample 5: A piece of wire measuring 20 feet is attached to a telephone pole as a guy wire. Step 6: Set each factor equal to 0. And solve the linear equation. Step 4: Write the equation in standard form. Substitute the given information to the equation. Step 3: Determine if there is a special formula needed. Step 1: Draw and label a picture if necessary. Įxample 1: A vacant rectangular lot is being turned into a community vegetable garden measuring 8 meters by 12 meters. A path of uniform width is to surround garden. If the area of the lot is 140 square meters, find the width of the path surrounding the garden. of carpet.)Īrea of a rectangle and Landscaping/border/frame problems. Set each factor equal to 0. And solve the linear equation. Substitute the given information into the equation.Ħ. Determine if there is a special formula needed. Steps for solving Quadratic application problems:ġ. ![]()
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