Name: Period: 1 Review Test 2: Quadratic Solving and Graphing

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1 Name: Period: Review Test : Quadratic Solving and Graphing Solving quadratics by Factoring A. Factoring Quadratics Examples of monomials: Examples of binomials: Examples of trinomials: Strategies to use: () Look for a GCF to factor out of all terms () Look for special factoring patterns as listed below () Use the X-Box method or the Grouping method (4) Check your factoring by using multiplication/foil Factor each expression completely. Check using multiplication..) x 5x.) x 4.) x 5x 4 4.) 5x 8 5.) m m+.) 4x + x+

2 .) + 8.) 5x x x 5x +.) 5t 0t + 0.) x.) a 4a ) x + x+ B. Solving quadratics using factoring To solve a quadratic equation is to find the x values for which the function is equal to. The solutions are called the or of the equation. To do this, we use the Zero Product Property: Zero Product Property List some pairs of numbers that multiply to zero: ( )( ) = 0 ( )( ) = 0 ( )( ) = 0 ( )( ) = 0 What did you notice? ZERO PRODUCT PROPERTY If the of two expressions is zero, then or of the expressions equals zero. Algebra If A and B are expressions and AB =, then A = or B =. Example If (x + 5)(x + ) = 0, then x + 5 = 0 or x + = 0. That is, or. Use this pattern to solve for the variable:. get the quadratic = 0 and factor completely. set each ( ) = 0 (this means to write two new equations). solve for the variable (you sometimes get more than solution)

3 Find the roots of each equation:.) x + x 0 = 0 4.) 4 8 x x+ = ) x x= Find the zeros of each equation:.) x + 8x 0 = x 4.) v(v + ) = 0 8.) x + x= 5 Find the zeros of the function by rewriting the function in intercept form:.) y = x x 0.) f ( x) x x g x = +.) ( ) x = 4 Converting between forms: From intercept form to standard form Use FOIL to multiply the binomials together Distribute the coefficient to all terms Ex: y = ( x+ 5)( x 8) From vertex form to standard form Re-write the squared term as the product of two binomials Use FOIL to multiply the binomials together Distribute the coefficient to all terms Add constant at the end Ex: ( ) ( ) f 4 x +

4 Graph the function. Label the vertex and axis of symmetry: y = x ( x+ ).) ( ) y x Vertex: Maximum or minimum value: x-intercepts: Axis of symmetry: Compare width to the graph of y = x.) k(x) = x x y x Vertex: Max or min? Direction of opening? Axis of symmetry: Compare to the graph of y = x 4.) h(x) = x + 4x + y x Vertex: Max or min? Direction of opening? Axis of symmetry: Compare to the graph of y = x

5 Solve Quadratic Equations by Finding Square Roots C. Simplifying Square Roots: Make a factor tree; circle pairs of buddies. One of each pair comes out of the root, the non-paired numbers stay in the root. Multiply the terms on the outside together; multiply the terms on the inside together Simplify:.) 0.) 5 D. Multiplying Square Roots: Simplify each radical completely by taking out buddies (outside outside) inside inside or ( a b)( c d ) = Simplify your answer, if possible Simplify: 5 5.) ( 5 )( 5 ) 4.) ( )( ) E. Simplifying Square Roots in Fractions: a Split up the fraction: b = 5 = 5 = 5 Simplify first by taking out buddies or reducing (you can only reduce two numbers that are both under a root or two numbers that are both not in a root) Square root top, square root bottom If one square root is left in the denominator, multiply the top and the bottom by the square root and simplify Reduce if possible Simplify:.).) 5.)

6 F. Solving Quadratic Equations Using Square Roots Isolate the variable or expression being squared (get it ) Square root both sides of the equation (include + and on the right side!) This means you have equations to solve!! Solve for the variable (make sure there are no roots in the denominator) 8.) x = 5.) x = 8 0.) 4x = 0.) m = 5.) (y + ) = 4.) ( x ) = 4 Solving quadratics is to use the quadratic formula. Solving using the quadratic formula: Put into standard form (ax + bx + c = 0) List a =, b =, c = b ± b 4ac Plug a, b, and c into a Simplify all roots (look for = i ); reduce

7 G. Solve by using the quadratic formula:.) x + (std. form): a = b = c = ± a b b 4ac.) 5x 8 - (std. form): a = b = c = ± b b 4ac a.) -x ) x = x 5.) -x ) 4( x ) = x+

8 Using the Discriminant Quadratic equations can have two, one, or no solutions (x-intercepts). You can determine how many solutions a quadratic equation has before you solve it by using the. The discriminant is the expression under the radical in the quadratic formula: Discriminant = b 4ac If b 4ac < 0, then the equation has imaginary solutions If b 4ac = 0, then the equation has real solution If b 4ac > 0, then the equation has real solutions ± 4 b b ac a H. Finding the number of x-intercepts Determine whether the graphs intersect the x-axis in zero, one, or two points..) y = 4x x+.) y = x x 0 I. Finding the number and type of solutions Find the discriminant of the quadratic equation and give the number and type of solutions of the equation..) x 5x= 4.) x 5.) x.) 4x = 5x +

9 J. Graph each function by making a table of values with at least 5 points. (A) State the vertex. (B) State the direction of opening (up/down). (C) State whether the graph is wider, narrower, or the same width as y = x. f( x) = ( x+ ) 5.) y x ) k(x) = x + x + y x Vertex: Direction of opening? Compare width to the graph of y = x Vertex: Direction of opening? Compare width to the graph of y = x K. Evaluate the following expressions given the functions below: g(x) = -5x - f(x) = x + 8 h( x) = j ( x) = x + 8 x a. g(8) = b. f(-4) = c. h( 5) = d. j( ) = e. g(4a + ) f. f(h(-)) h. Find x if g(x) = 4 i. Find x if h(x) = j. Find x if g(j(x)) = -

10 Methods for Solving Quadratic Equations: SUMMARY A.) Factoring st : Set equal to 0 nd : Factor out the GCF rd : Complete the X & box method to find the factors 4 th : Set every factor that contains an x in it, equal to 0 & solve for x. B.) Completing the Square st : Move the constant (number with no variable) to the right so that all variables are on the left & all constants are on the right. nd : Divide every term in the equation by the value of a, if it is not already. rd b : Create a perfect square trinomial on the left side by adding to both sides. 4 th b : Factor the left side into a x ± and simplify the value on the right side. 5 th : Take the square root of both sides of the equation. REMINDER: Don t forget the ± th : Solve for x C.) Finding Square Roots st : Isolate the term with the square. nd : Take the square root of both sides of the equation. REMINDER: Don t forget the ± rd : Solve for x. D.) Quadratic Formula st : Set the equation equal to 0. nd : Find the values of a, b, and c & plug them into the Quadratic Formula: b ± b 4ac a rd : Simplify the radical as much as possible. 4 th : If possible, simplify the numerator into integers. 5 th : Divide. REMINDER: If you have terms in the numerator (ex: 4 ± ), divide BOTH terms by the number in the denominator (the example would result in ± ) WORKED OUT QUADRATIC PROBLEMS Real-World Application Use the following problems to have students determine the best method to use to solve them. A sprinkler sprays water that covers a circular region of A= πr 5π = πr 5= r 5π square feet. Find the diameter of the circle. ± 5 = r ± 5 = r Since a diameter cannot be negative, use the positive square root. The diameter is twice the radius. The diameter of the circle is 0 ft.

11 The length of a rectangle is three more than twice the width. Determine the dimensio a total area of m. = 0 = w = ( w + )( w) = w + w 0 = w + w ( w + )( w ) 0 = w + and and 0 = w w = Since w represents the width of a rectangle we must omit the negative value. Therefore, we have w =. The dimensions that give the rectangle an area of m are m by m. Ways to Solve Quadratic Equations and When to Use: Solving by Factoring Use When... Example # Example # You are told to solve by factoring. x 4x = 0 x + x + = 0 The quadratic is easily factorable. ( x )( x + ) = 0 ( x + )( x + ) = 0 The quadratic is already x = 0 x + = 0 x + = 0 x + = 0 factored, such as: ( x + )( x + ) = 0 Some quadratic equations can be solved by factoring. Set the equation equal to zero and factor. The discriminant is positive and a perfect square. Solving by Graphing Use When... Example # x Example # x You are told to solve by graphing. and.5 and The graph easily shows integer values for the x-intercepts. An approximate solution is sufficient. The discriminant is positive and/or = 0. Set the equation equal to zero, if necessary. Find the roots using the ZERO command tool of the graphing calculator. Graph each side of the equation separately. Use the INTERSECT command tool to find when the graphs cross at both points.

12 Solving by Square Roots Use When... Example # Example # You are told to solve by the square root method. ( x + ) = ( x + ) = x is set equal to a numeric value, such as: x + = ± ( x + ) = 4 ± x = ± ( x + ) = 8 The middle term, bx, is missing, such as: x + = ± 8 ± x 5 = 0 When using this method, you will have to make sure students use the ± which ties back to the quadratic formula and why there should be two answers. You have the difference of two squares, such as: x 4 = 0 The discriminant is positive and/or = 0. Solving by Quadratic Formula Example # x 5 = 0 a =, b = 0, c = -5 b ± b 4 ac a ± 4( )( 5) ± 0 ± 5 ( ) ± 5 Example # x x 4 = 0 a =, b = -, c = -4 b ± ± b 4 ac a ± 80 8 ( ) 4( )( 4) ( ) ( ± 5) 8 ± 5 8 ± 5 The solutions of some quadratic equations are not rational numbers and cannot be factored. The equation should be set equal to zero before using the formula. Use When... You are told to use the quadratic formula. Factoring looks difficult, or you are having trouble finding the correct factors. 0x x 4 = 0 The quadratic is not factorable, such as: x 5 = 0 The question asks for the answer to be rounded. Solving any quadratic equation. Solving by Completing the Square Example # x 8x = 5 x 8x + = 5+ = 4 ( x 4) x 4 = ± 4 ± 4 4 Example # x x x x + x = 5 + x + = 5+ ( x + ) + x 5= 0 + x = 5 = x + = ± Use When... You are told to solve by completing the square. Have to put the quadratic in vertex form, such as: a ( x h) + k = 0 a is and b is even The discriminant is positive and/or = 0. ± b When completing the square to solve an equation, be sure to add coefficient of the x term is. to each side. Before you complete the square, be sure that the