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Solving quadratic equations by factoring Quadratic Equation Standard form: , Where a does not equal 0. Example 1 Solve by factoring. Step 1: Simplify each side if needed. This quadratic equation is already simplified. Step 2: Write in standard form, , if needed. This quadratic equation is already in standard form. Step 3: Factor. Step 4: Use the Zero-Product Principle AND Step 5: Solve for the linear equation(s) set up in step 4. 1 *Use Zero-Product Principle *Solve the first linear equation *Solve the second linear equation There are two solutions to this quadratic equation: x = -5 and x = 2. Example 2: Solve by factoring. Step 1: Simplify each side if needed. *Mult. both sides by LCD of 6 to clear fractions Step 2: Write in standard form, , if needed. 2 *Inverse of add. 16 is sub. 16 *Quad. eq. in standard form Step 3: Factor. *Quad. eq. in standard form *Factor the diff. of two squares Step 4: Use the Zero-Product Principle AND Step 5: Solve for the linear equation(s) set up in step 4. *Use Zero-Product Principle *Solve the first linear equation *Solve the second linear equation There are two solutions to this quadratic equation: x = -4/5 and x = 4/5. 3 Example 3: Solve by factoring. Step 1: Simplify each side if needed. *Use Dist. Prop. to clear the ( ) Step 2: Write in standard form, , if needed. *Inverse of add. 2 is sub. 2 *Quad. eq. in standard form Step 3: Factor. *Quad. eq. in standard form *Factor the trinomial Step 4: Use the Zero-Product Principle AND Step 5: Solve for the linear equation(s) set up in step 4. 4 *Use Zero-Product Principle *Solve the first linear equation *Solve the second linear equation There are two solutions to this quadratic equation: x = -2/3 and x = 1/2. 5 Solving Quadratic Equations by the Square Root Method You can solve a quadratic equation by the square root method if you can write it in the form . Step 1: Write the quadratic equation in the form if needed. A and B represent algebraic expressions. When you have the quadratic equation written in this form, it allows you to use the square root method described in step 2. If it is not in this form, , move any term(s) to the appropriate side by using the addition/subtraction or multiplication/division property of equality. Step 2: Apply the square root method. If A and B are algebraic expressions such that , then also written , . In other words, if you have an expression squared set equal to another expression, the inverse operation to solve it is to take the square root of both sides. Since both a positive and its opposite squared result in the same answer, then you will have two answers, plus or minus the square root of B. Step 3: Solve for the linear equation(s) set up in step 2. After applying the square root method to a quadratic equation you will end up with either one or two linear equations to solve. Most times you will have two linear equations, but if B is equal to 0, then you will only have one since plus or minus 0 is only one number. 6 Example 4: Solve by using the square root method. Step 1: Write the quadratic equation in the form if needed AND Step 2: Apply the square root method. *Written in the form *Apply the sq. root method *There are 2 solutions Step 3: Solve for the linear equation(s) set up in step 2. *Sq. root of 16 = 4 *Neg. sq. root of 16 = - 4 There are two solutions to this quadratic equation: x = 4 and x = 4. Example 5: Solve by using the square root method. 7 Step 1: Write the quadratic equation in the form if needed AND Step 2: Apply the square root method. Note how this quadratic equation is not in the form to begin with. The 5 is NOT part of the expression being squared on the left by side of the equation. We can easily write it in the form dividing both sides by 5. *Not in the form *Inv. of mult. by 5 is div. by 5 *Written in the form *Apply the sq. root method *There are 2 solutions Step 3: Solve for the linear equation(s) set up in step 2. *Sq. root of 4 = 2 *Neg. sq. root of 4 = -2 There are two solutions to this quadratic equation: x = 2 and x = 2. 8 Example 6: Solve by using the square root method. Step 1: Write the quadratic equation in the form if needed AND Step 2: Apply the square root method. *Written in the form *Apply the sq. root method *There are 2 solutions Step 3: Solve for the linear equation(s) set up in step 2. 9 *Sq. root of 20 = 2 sq. root of 5 *Solve for x *Neg. sq. root of 20 = -2 sq. root of 5 *Solve for x There are two solutions to this quadratic equation: x = and x = . 10 Solving Quadratic Equations by Completing the Square You can solve ANY quadratic equation by completing the square. This comes in handy when a quadratic equation does not factor or is difficult to factor. Step 1:Make sure that the coefficient on the term is equal to 1. If the coefficient of the term is already 1, then proceed to step 2. If the coefficient of the by that coefficient. term is not equal to 1, then divide both sides Step 2:Isolate the and x terms. In other words, rewrite it so that the the constant is on the other side. and x terms are on one side and Step 3: Complete the square. At this point we will be creating a perfect square trinomial (PST). Recall that a PST is a trinomial of the form and it factors . When it is in that form it will allow us to in the form continue onto the next step and take the square root of both sides and find a solution. We need to find a number that we can add to the that we have a PST. and x terms so We can get that magic number by doing the following: If we have we can complete it’s square by adding the 11 constant In other words, we complete the square by taking ½ of b (the coefficient of the x term) and then squaring it. Make sure you remember to add it to BOTH sides to keep the equation balanced. Step 4: Factor the perfect square trinomial (created in step 3) as a binomial squared. If you need a review on factoring a perfect square trinomial, feel free to go to Tutorial 7: Factoring Polynomials. Step 5: Solve the equation in step 4 by using the square root method. Example 7: Solve by completing the square. Step 1:Make sure that the coefficient on the The coefficient of the Step 2: Isolate the The term is equal to 1. term is already 1. and x terms. and x terms are already isolated. Step 3: Complete the square. 12 *b is the coefficient of the x term *Complete the square by taking 1/2 of b and squaring it *Add constant found above to BOTH sides of the eq. *This creates a PST on the left side of eq. Step 4: Factor the perfect square trinomial (created in step 3) as a binomial squared. *Factor the PST Step 5: Solve the equation in step 4 by using the square root method. *Written in the form *Apply the sq. root method *There are 2 solutions 13 There are two solutions to this quadratic equation: x = 9 and x = 1. Example 8: Solve by completing the square. Step 1:Make sure that the coefficient on the term is equal to 1. term is not 1 to begin with. We can Note how the coefficient on the easily fix that by dividing both sides by that coefficient, which in this case is 3 . *Divide both sides by 3 *Coefficient of Step 2: Isolate the term is now 1 and x terms. and x terms are not isolated to begin with. We can Note how the easily fix that by moving the constant to the other side of the equation. *Inverse of add. 3 is sub. 3 * and x terms are now isolated Step 3: Complete the square. 14 *b is the coefficient of the x term *Complete the square by taking 1/2 of b and squaring it *Add constant found above to BOTH sides of the eq. *This creates a PST on the left side of eq. Step 4: Factor the perfect square trinomial (created in step 3) as a binomial squared. *Factor the PST Step 5: Solve the equation in step 4 by using the square root method. 15 *Written in the form *Apply the sq. root method *There are 2 solutions *Square root of a negative 1 is i *Square root of a negative 1 is i There are two solutions to this quadratic equation: x = and x = . 16