{"id":19666,"date":"2022-01-02T10:00:40","date_gmt":"2022-01-02T04:30:40","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=19666"},"modified":"2023-01-03T09:48:47","modified_gmt":"2023-01-03T04:18:47","slug":"math-labs-activity-interpret-geometrically-factors-quadratic-expression","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/math-labs-activity-interpret-geometrically-factors-quadratic-expression\/","title":{"rendered":"Math Labs with Activity – Interpret Geometrically the Factors of a Quadratic Expression"},"content":{"rendered":"

Math Labs with Activity – Interpret Geometrically the Factors of a Quadratic Expression<\/strong><\/span><\/h2>\n

OBJECTIVE<\/strong><\/span><\/p>\n

To interpret geometrically the factors of a quadratic expression of the form x2<\/sup> + bx + c<\/p>\n

Materials Required<\/strong><\/span><\/p>\n

    \n
  1. Two sheets of graph paper<\/li>\n
  2. A geometry box<\/li>\n<\/ol>\n

    Theory<\/strong> <\/span>
    \nThe polynomials (x + p) and (x + q) are the factors of a
    \nquadratic expression x2<\/sup>+bx + c if p + q=b and pq = c.<\/p>\n

    Procedure<\/strong> <\/span>
    \nFor convenience, we take x = 10.
    \nCase I –<\/strong> Let us first consider the expression x2<\/sup> +7x+ 12. Then, b =7 and c = 12
    \nStep 1:<\/strong> Find two numbers such that their sum is 7 and their product is 12. Two such numbers are 3 and 4.
    \nStep 2:<\/strong> Construct a square ABCD having each side = x (= 10 cm) on a graph paper as shown in Figure 12.1.
    \nStep 3:<\/strong> Construct a rectangle BEFC having sides BE=CF = 3 cm and BC =EF = x (= 10 cm) as shown in Figure 12.1. Shade the rectangle BEFC.
    \nStep 4:<\/strong> Construct a rectangle DCGH having sides DC =GH = x (= 10 cm) and DH =CG =4 cm as shown in Figure 12.1. Shade the rectangle DCGH.
    \nStep 5:<\/strong> Construct a rectangle CFIG having sides CF =GI = 3 cm and CG = FI = 4 cm as shown in Figure
    \nStep 6:<\/strong> Record your observations (see Observations and Calculations, Case I).
    \n\"Math
    \nCase II –<\/strong> Let us now consider the expression x2<\/sup> -x -12 Then, b = -1 and c = -12
    \nStep 1:<\/strong> Find two numbers such that their sum is -1 and their product is -12.
    \nTwo such numbers are -4 and +3.
    \nStep 2:<\/strong> Construct a square KLMN having each side = x (= 10 cm) on the second graph paper as shown in Figure 12.2.
    \nStep 3:<\/strong> Construct a rectangle QLMP (inside the square KLMN) having sides QL = PM = 4 cm and sides LM=QP = x (= 10 cm) as shown in Figure 12.2. Shade the rectangle QLMP.
    \nStep 4:<\/strong> Construct a rectangle NMRS having sides NM = SR=x (= 10 cm) and NS=MR = 3 cm as shown in Figure 12.2.
    \nStep 5:<\/strong> Mark a point T on the line SR so as to get a rectangle PMRT having sides PM=TR= 4 cm and FT = MR = 3 cm as shown in Figure 12.2.
    \nStep 6:<\/strong> Record your observations (see Observations and Calculations, Case II).
    \n\"Math<\/p>\n

    Observations and Calculations<\/strong><\/span>
    \nCase I –<\/strong> In Figure 12.1, we have<\/p>\n

      \n
    1. Area of the rectangle AEIH=(x + 3)(x+4) cm2<\/sup>.<\/li>\n
    2. (a)<\/strong> Area of the rectangle ABCD = (x2<\/sup>) cm2<\/sup>.
      \n(b)<\/strong> Area of the rectangle BEFC = (3x) cm2<\/sup>.
      \n(c)<\/strong> Area of the rectangle DCGH = (4x) cm2<\/sup>.
      \n(d)<\/strong> Area of the rectangle CFIG = 3 x 4 = 12 cm2<\/sup>.
      \narea of the rectangle AEIH=(x2<\/sup> + 3x + 4x+12) cm2<\/sup>. From (i) and (ii) we get
      \n(x + 3)(x + 4) =x2<\/sup> + 3x + 4x +12 => x\u00b2 +7x + 12 = (x+3)(x+4).
      \n\u2234 (x + 3)and(x + 4)are the two factors of x2<\/sup>+7x + 12.<\/li>\n<\/ol>\n

      Case II –<\/strong> In Figure 12.2, we have<\/p>\n

        \n
      1. Area of the rectangle KQTS = (x-4)(x + 3) cm2<\/sup>.<\/li>\n
      2. (a)<\/strong> Area of the square KLMN=(x2<\/sup>) cm2<\/sup>.
        \n(b)<\/strong> Area of the rectangle QLMP =(4x) cm2<\/sup>.
        \n(c)<\/strong> Area of the rectangle NMRS = (3x) cm2<\/sup>.
        \n(d)<\/strong> Area of the rectangle PMRT = 3 x 4 = 12 cm2<\/sup>. area of the rectangle KQTS = [area (square KLMN) – area (rect. QLMP) + area (rect. NMRS) – area (rect. PMRT)]
        \n=(x2<\/sup>-4x+3x-12) cm2<\/sup>.
        \nFrom (i) and (ii) we get
        \n(x-4)(x+3) =x2<\/sup>-4x+3x-12 => x2<\/sup>-x-12 = (x-4)(x + 3).
        \n\u2234 (x-4) and (x + 3) are the two factors of x2<\/sup> -x-12.<\/li>\n<\/ol>\n

        Result<\/strong> <\/span>
        \nThe method discussed above gives the geometrical interpretation of the factorisation of a quadratic expression of the form\u00a0x2<\/sup> +bx + c.<\/p>\n

        Remarks:<\/strong> The teacher must ask the students to factorise other quadratic expressions of the form\u00a0x2<\/sup> +bx + c. The value of x may also be taken to be other than 10 cm.<\/p>\n

        Math Labs with Activity<\/a>Math Labs<\/a>Math Lab Manual<\/a>Science Labs<\/a>Science Practical Skills<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

        Math Labs with Activity – Interpret Geometrically the Factors of a Quadratic Expression OBJECTIVE To interpret geometrically the factors of a quadratic expression of the form x2 + bx + c Materials Required Two sheets of graph paper A geometry box Theory The polynomials (x + p) and (x + q) are the factors of […]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[6805],"tags":[],"yoast_head":"\nMath Labs with Activity - Interpret Geometrically the Factors of a Quadratic Expression - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/math-labs-activity-interpret-geometrically-factors-quadratic-expression\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Math Labs with Activity - 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