**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 x^{2} + bx + c

**Materials Required**

- Two sheets of graph paper
- A geometry box

**Theory**

The polynomials (x + p) and (x + q) are the factors of a

quadratic expression x^{2}+bx + c if p + q=b and pq = c.

**Procedure**

For convenience, we take x = 10.

**Case I –** Let us first consider the expression x^{2} +7x+ 12. Then, b =7 and c = 12

**Step 1:** Find two numbers such that their sum is 7 and their product is 12. Two such numbers are 3 and 4.

**Step 2:** Construct a square ABCD having each side = x (= 10 cm) on a graph paper as shown in Figure 12.1.

**Step 3:** 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.

**Step 4:** 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.

**Step 5:** Construct a rectangle CFIG having sides CF =GI = 3 cm and CG = FI = 4 cm as shown in Figure

**Step 6:** Record your observations (see Observations and Calculations, Case I).

**Case II –** Let us now consider the expression x^{2} -x -12 Then, b = -1 and c = -12

**Step 1:** Find two numbers such that their sum is -1 and their product is -12.

Two such numbers are -4 and +3.

**Step 2:** Construct a square KLMN having each side = x (= 10 cm) on the second graph paper as shown in Figure 12.2.

**Step 3:** 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.

**Step 4:** Construct a rectangle NMRS having sides NM = SR=x (= 10 cm) and NS=MR = 3 cm as shown in Figure 12.2.

**Step 5:** 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.

**Step 6:** Record your observations (see Observations and Calculations, Case II).

**Observations and Calculations**

**Case I –** In Figure 12.1, we have

- Area of the rectangle AEIH=(x + 3)(x+4) cm
^{2}. **(a)**Area of the rectangle ABCD = (x^{2}) cm^{2}.

**(b)**Area of the rectangle BEFC = (3x) cm^{2}.

**(c)**Area of the rectangle DCGH = (4x) cm^{2}.

**(d)**Area of the rectangle CFIG = 3 x 4 = 12 cm^{2}.

area of the rectangle AEIH=(x^{2}+ 3x + 4x+12) cm^{2}. From (i) and (ii) we get

(x + 3)(x + 4) =x^{2}+ 3x + 4x +12 => x² +7x + 12 = (x+3)(x+4).

∴ (x + 3)and(x + 4)are the two factors of x^{2}+7x + 12.

**Case II –** In Figure 12.2, we have

- Area of the rectangle KQTS = (x-4)(x + 3) cm
^{2}. **(a)**Area of the square KLMN=(x^{2}) cm^{2}.

**(b)**Area of the rectangle QLMP =(4x) cm^{2}.

**(c)**Area of the rectangle NMRS = (3x) cm^{2}.

**(d)**Area of the rectangle PMRT = 3 x 4 = 12 cm^{2}. area of the rectangle KQTS = [area (square KLMN) – area (rect. QLMP) + area (rect. NMRS) – area (rect. PMRT)]

=(x^{2}-4x+3x-12) cm^{2}.

From (i) and (ii) we get

(x-4)(x+3) =x^{2}-4x+3x-12 => x^{2}-x-12 = (x-4)(x + 3).

∴ (x-4) and (x + 3) are the two factors of x^{2}-x-12.

**Result**

The method discussed above gives the geometrical interpretation of the factorisation of a quadratic expression of the form x^{2} +bx + c.

**Remarks:** The teacher must ask the students to factorise other quadratic expressions of the form x^{2} +bx + c. The value of x may also be taken to be other than 10 cm.

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