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## Hawkes Learning Systems: College Algebra

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**Hawkes Learning Systems:College Algebra**Section 2.3: Quadratic Equations in One Variable**Objectives**• Solving quadratic equations by factoring. • Solving “perfect square” quadratic equations. • Solving quadratic equations by completing the square. • The quadratic formula. • Applications: gravity problems.**Quadratic Equations**A quadratic equation in one variable, say the variable is an equation that can be transformed into the form Where , , and are real numbers and . Such equations are also called second-degree equations, as appears to the second power. The name quadratic comes from the Latin word quadrus, meaning “square”.**Solving Quadratic Equations by Factoring**Let and represent algebraic expressions. If the product of and is , then at least one of and itself is . That is, For example, Zero –Factor Property**Solving Quadratic Equations by Factoring**• The key to using factoring to solve a quadratic equation is to rewrite the equation so that appears by itself on one side of the equation. • If the trinomial can be factored, it can be written as a product of two linear factors and . • The Zero-Factor Property then implies that the only way for to be is if one (or both) of and is .**Example: Solving Quadratic Equations by Factoring**Solve the quadratic equation by factoring. Step 1: Multiply both sides by . Step 2: Subtract from both sides so is on one side. Step 3: Factor and solve the two linear equations. or or**Example: Solving Quadratic Equations by Factoring**Solve the quadratic equation by factoring. or In this example, the two linear factors are the same. In such cases, the single solution is called a double solution or a double root.**Example: Solving Quadratic Equations by Factoring**Solve the quadratic equation by factoring. or or**Solving “Perfect Square” Quadratic Equations**In some cases where the factoring method is unsuitable, the solution to a second-degree polynomial can be obtained by using our knowledge of square roots. Ifis an algebraic expression and if is a constant: If a given quadratic equation can be written in the form we can use the above observation to obtain two linear equations that can be easily solved. implies**Example: Square Root Method**Solve the quadratic equation using the Square Root Method. a) x2= 25 b) 6x2= 54**Example: “Perfect Square” Quadratic Equations**Solve the quadratic equation by taking square roots. Step 1: Take the square root of each side. Step 2: Subtract from both sides. Step 3: Divide both sides by .**Example: “Perfect Square” Quadratic Equations**Solve the quadratic equation by taking square roots. In this example, taking square roots leads to two complex number solutions.**Solving Quadratic Equations by Completing the Square**Method of Completing the Square Step 1: Write the equation in the form . Step 2: Divide byif, so the coefficient of is : Step 3: Divide the coefficient of by , square the result, and add this to both sides. Step 4: The trinomial on the left side will now be a perfect square. That is, it can be written as the square of an algebraic expression.**Example: Completing the Square**Solve the quadratic equation by completing the square. Step 1: Move the constant term to the other side of the equation. Step 2: Divide by , the coefficient of x2 . Step 3: Add to both sides. Step 4: Factor the trinomial, and solve.**The Quadratic Formula**The method of completing the square can be used to derive the quadratic formula,a formula that gives the solution to any equation of the form . Step 1: Move the constant to the other side of the equation. Step 2: Divide by . Step 3: Divide by square the result, and add it to both sides of the equation.**The Quadratic Formula**Deriving the quadratic formula (cont.). Step 4: Factor the trinomial, and solve for .**The Quadratic Formula**The solutions of the equation are: Note: • The equation has 2 real solutions if . • The equation has 1 real solution if . • The equation has 2 complex solutions (which are conjugates of one another) if .**Example: The Quadratic Formula**Solve using the quadratic formula.**Gravity Problems**• When an object near the surface of the Earth is moving under the influence of gravity alone, its height above the surface of the earth is described by a quadratic polynomial in the variable where stands for time and is usually measured in seconds. • In some applications involving this formula, one of the two solutions must be discarded as meaningless in the given problem.**Gravity Problems**If we let represent the height at time , where , , and are all constants: • g = acceleration due to gravity (32 ft/sec2or 9.8 m/sec2)**Example: Gravity Problems**Luke stands on a tier of seats in a baseball stadium, and throws a ball out onto the field with a vertical upward velocity of . The ball is above the ground at the moment he releases it. When does the ball land? Since we know the answer cannot be negative, this solution is discarded.