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Chapter 5: Problem 4

Factorise (a) \(x^{2}+8 x-9\), (b) \(x^{2}+9 x-22\) (c) \(x^{2}+10 x+9\), (d) \(x^{2}+7 x+12\) (e) \(x^{2}-7 x+12\)

Short Answer

Expert verified
Question: Factorise the following quadratic expressions: 1. \(x^{2} + 8x - 9\) 2. \(x^{2} + 9x - 22\) 3. \(x^{2} + 10x + 9\) 4. \(x^{2} + 7x + 12\) 5. \(x^{2} - 7x + 12\) Answer: 1. \(x^{2} + 8x - 9 = (x + 9)(x -1)\). 2. \(x^{2} + 9x - 22 = (x + 11)(x - 2)\). 3. \(x^{2} + 10x + 9 = (x + 9)(x + 1)\). 4. \(x^{2} + 7x + 12 = (x + 4)(x + 3)\). 5. \(x^{2} - 7x + 12 = (x - 4)(x - 3)\).

Step by step solution

01

Factorise \(x^{2} + 8x - 9\)#

Step 1: Identify the quadratic in the form \(ax^2 + bx + c\). Here, \(a = 1, b = 8,\) and \(c = -9\). Step 2: Find two numbers that multiply to \(c\) and add to \(b\). We are looking for two numbers that multiply to \(-9\) and add to \(8\). These numbers are \(9\) and \(-1\). Step 3: Write the binomials using the numbers found. Since the coefficients of \(x^2\) are \(1\), we can simply write the two binomials as \((x + 9)(x -1)\). So, the factorisation is: \(x^{2} + 8x - 9 = (x + 9)(x -1)\).
02

Factorise \(x^{2} + 9x - 22\)#

Step 1: Identify the quadratic in the form \(ax^2 + bx + c\). Here, \(a = 1, b = 9,\) and \(c = -22\). Step 2: Find two numbers that multiply to \(c\) and add to \(b\). We are looking for two numbers that multiply to \(-22\) and add to \(9\). These numbers are \(11\) and \(-2\). Step 3: Write the binomials using the numbers found. The two binomials are \((x + 11)(x - 2)\). So, the factorisation is: \(x^{2} + 9x - 22 = (x + 11)(x - 2)\).
03

Factorise \(x^{2} + 10x + 9\)#

Step 1: Identify the quadratic in the form \(ax^2 + bx + c\). Here, \(a = 1, b = 10,\) and \(c = 9\). Step 2: Find two numbers that multiply to \(c\) and add to \(b\). We are looking for two numbers that multiply to \(9\) and add to \(10\). These numbers are \(9\) and \(1\). Step 3: Write the binomials using the numbers found. The two binomials are \((x + 9)(x + 1)\). So, the factorisation is: \(x^{2} + 10x + 9 = (x + 9)(x + 1)\).
04

Factorise \(x^{2} + 7x + 12\)#

Step 1: Identify the quadratic in the form \(ax^2 + bx + c\). Here, \(a = 1, b = 7,\) and \(c = 12\). Step 2: Find two numbers that multiply to \(c\) and add to \(b\). We are looking for two numbers that multiply to \(12\) and add to \(7\). These numbers are \(4\) and \(3\). Step 3: Write the binomials using the numbers found. The two binomials are \((x + 4)(x + 3)\). So, the factorisation is: \(x^{2} + 7x + 12 = (x + 4)(x + 3)\).
05

Factorise \(x^{2} - 7x + 12\)#

Step 1: Identify the quadratic in the form \(ax^2 + bx + c\). Here, \(a = 1, b = -7,\) and \(c = 12\). Step 2: Find two numbers that multiply to \(c\) and add to \(b\). We are looking for two numbers that multiply to \(12\) and add to \(-7\). These numbers are \(-4\) and \(-3\). Step 3: Write the binomials using the numbers found. The two binomials are \((x - 4)(x - 3)\). So, the factorisation is: \(x^{2} - 7x + 12 = (x - 4)(x - 3)\).

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