5.2 Solving Quadratic Equations by Factoring

Transcription

1 Name. Solving Quadratic Equations b Factoring MATHPOWER TM, Ontario Edition, pp To solve a quadratic equation b factoring, a) write the equation in the form a + b + c = b) factor a + b + c c) use the zero product propert d) solve the two resulting equations to find the roots e) check our solutions. State the roots of each equation. a) ( )( + 7) = b) ( + )( ) = c) 7( ) = d) ( + )( + ) = Problem Solving. The hpotenuse of a right triangle is m longer than twice one of the other two sides. The third side of the triangle is m. Find the lengths of the unknown side and hpotenuse.. Write each equation in the form a + b + c =. a) = ( ) b) ( ) = + 7 c) = d) =. The length of a flag is twice the width. Find the width of a flag with an area of cm.. Solve and check. a) + 8 = b) + + =. Three consecutive even integers are such that the product of the first two is less than 9 times the third. Find the integers. c) q q 8 = d) m + m 7 = e) t 7t + = f) = g) t t = h) p p = 7. A rectangle is cm long and cm wide. When each dimension is increased b the same amount, the area is doubled. What are the new dimensions? i) ( ) = 7 + j) a a = k) ( ) = l) =. 8. Communication Write a quadratic equation whose roots are and Eplain our. thinking. Chapter Copright McGraw-Hill Rerson Limited

2 Name. Investigation: Graphing Quadratic Functions b Factoring MATHPOWER TM, Ontario Edition, p. 87 To sketch the graph of a quadratic function, a) find the -intercepts b factoring the equation in the form a + b + c = b) plot the points where the graph crosses the -ais, ( = ) c) use smmetr to find the -coordinate of the verte d) find the -coordinate of the verte b substituting the -coordinate of the verte into the equation in the form = a + b + c e) draw a smooth curve through the three points. Sketch the graphs of the following quadratic functions b locating the -intercepts, and then finding the coordinates of the verte. a) = ( )( ) b) = e) = + f) = 8. Use the -intercepts to determine the coordinates of the points on the -ais and the verte for the graph of each quadratic function. a) = ( + )( ) b) = 9 g) = + h) = + +. Sketch the graphs of the following quadratic functions b factoring to find the -intercepts, and then deducing the coordinates of the verte. a) = + b) = 7 +. Communication Eplain how to sketch a graph of each function using the intercepts and ais of smmetr. a) = ( ) c) = + d) =. b) = ( + ) Copright McGraw-Hill Rerson Limited Chapter

3 . The Quadratic Formula Name MATHPOWER TM, Ontario Edition, pp To solve a quadratic equation using the quadratic formula, write the equation in the form a + b + c =, a. b b ac The quadratic formula is = ±. a. Solve using the quadratic formula. a) 8 + = b) =. Application Use the Pthagorean Theorem to find the value of, to the nearest hundredth. + c) + 7 = d) =. Solve using the quadratic formula. Epress answers as eact roots. a) + = b) + + = 7 Problem Solving. The sum of the squares of three consecutive odd integers is 87. Find the integers. c) 8 = d) ( + 8) =. A plain mat is placed around a picture measuring 8 cm b cm so that the width of the mat is equal on all sides. The area of the mat is of the area of the picture. Find the width of the mat, to the nearest millimetre. e) b + b = f) + =. Solve using the quadratic formula. Round to the nearest hundredth, if necessar. a) = b) 9 = 7. A window is in the shape of a rectangle surmounted b a semicircle. The height of the rectangle is. m more than the width. The total area of the window is. m. Find the width and height of the window, to the nearest hundredth. c).( + ) =. d) = e) a = f) ( + ) + ( + ) = 8. Communication Is it possible to write two real numbers whose sum is and whose product is? Use the quadratic formula to help ou eplain. Chapter Copright McGraw-Hill Rerson Limited

4 Answers CHAPTER Quadratic Equations. Solving Quadratic Equations b Graphing. a) b) = 7 +. a), b), c), d), = + e). f)., g),. h) no solution i), j).,.. s. 8 m b m.,, or,, (, ) (, ) (, ) c) d) (, ) (, ) = + (, ) (, ) 7, (, 8). m b m b m =.. cm, 9 cm. Solving Quadratic Equations b Factoring. a), 7 b), c), d). a) + = b) 7 = c) = d) =. a), b), c), 7 d) 9, e) 7, f) g), h), 8 i) j), k),, l),. 8 m, 7 m. cm.,, 7. cm b cm 8. = and = so ( + ) Epand the =., equation to get + =. Then, multipl ever term b to eliminate the denominators, to get + = or + =.. Investigation: Graphing Quadratic Functions b Factoring. a) b) = ( )( ) (, ) (, ) (, ) (, ) (, ) = (, ) (, ) (, ) (, ) e) f) ( 7, ) (, ) = + (, ) g) h) (, ) (, ) (, ) = + (, ) (, ) = 8, = + + (, ) (, ),. a) The equation is in factored form, so there is onl one -intercept,. Plot (, ). To find the -intercept, let = and find. Plot (, ). A third point is smmetrical to (, ) about the ais of smmetr =. Plot (, ). Check b substituting (, ) into the equation.. a) (, ), (, ), (, ) b) ( 7, ), (7, ), (, 9) Copright McGraw-Hill Rerson Limited Chapter

5 b) There is one -intercept,. Plot (, ). To find the -intercept, let = and find. Plot (, ). A third point is smmetrical to (, ) about the ais of smmetr =. Plot (, ). Check b substituting (, ) into the equation.. The Quadratic Formula. a), b) c) 7,, d). a) ± b) ± c) d) ± e) f) no solution. a).7,.7 b).9,. c) 7.7,. d) no solution e) ±. f).9,.9. =.98., 7, 9, or 9, 7,.. cm or mm 7..9 m wide,.99 m high 8. If one number is and the other number is, the product of the two numbers, ( ), should equal. Tr to solve + =. There are no real solutions. ± 8± 7, 8 Chapter Copright McGraw-Hill Rerson Limited