**A**

*quadratic equation*is an equation which is of the form*ax*^{2}+*bx*+*c*= 0 where*a*,*b*, and*c*are real numbers and*a is not equal to zero*.**Since it is a second degree equation, it has two solutions. The solutions of a quadratic equation are called the roots of the quadratic equation.**

**We can use the following methods to solve quadratic equations:-**

**a) Solve Quadratic Equations by Factoring.**

**b) Solve Quadratic Equations by Finding Square Roots.**

**c) Solve Quadratic Equations by Completing the Square.**

**d) Solve Quadratic Equations using Quadratic Formula.**

**e) Solve Quadratic Equations by Graphing.**

**We can solve the quadratic equations by using factoring method.**

**The general form of a quadratic equation is,**

*x*^{2}- (sum of the roots) x + product of the roots = 0**· To factor a quadratic equation which is of the form,**

*x*^{2}+*bx*+*c,*find two real numbers*m*and*n*such that,

*mn*=*c*and*m*+*n*=*b***If we find two integers**

*m*and*n*with the above conditions, then we can write*x*^{2}+*bx*+*c*as (*x*+*m*)(*x*+*n*) and then use zero product property to find the solution.**· If the quadratic equation is of the form**

*ax*^{2}+*bx*+*c*, then find two real numbers*m*and*n*such that,

*mn*=*ac*and*m*+*n*=*b***If we find the two integers**

*m*and*n*with the above conditions, then we can write*ax*^{2}+*bx*+*c*as*ax*^{2}+*mx*+*nx*+*c*and then go ahead with factor by grouping method.**Let us solve a few examples to understand factoring method.**

**Example 1:**

**Solve**

*x*^{2}-*x*- 30 = 0 by factoring method.**Solution:**

**Step 1:**

**The given quadratic equation is of the form**

*x*^{2}+*bx*+*c*= 0, where*b*= -1 and*c*= -30.**Now, find two real numbers**

*m*and*n*such that,*m*+*n*= -1 and*mn*= -30.**The two numbers are 5 and -6.**

**Step 2:**

**Sum of the roots = 5 + (-6)**

**= 5 - 6**

**= -1**

**Step 3:**

**Product of the roots = (5)(-6)**

**= -30**

**Step 4:**

**So, we can write**

*x*^{2}-*x*- 30 as (*x*+ 5) and (*x*- 6).**(**

*x*+ 5)(*x*- 6) = 0**Now, use zero product property and simplify.**

**(**

*x*+ 5)(*x*- 6) = 0

*x*+ 5 = 0 or*x*- 6 = 0

*x*= -5 or*x*= 6**Step 5:**

**So, the solution set is {-5, 6}.**

**Example 2:**

**Solve 2**

*x*^{2}+*x*- 45 = 0 by factoring method.**Solution:**

**Step 1:**

**The given quadratic equation is 2**

*x*^{2}+*x*- 45 = 0, which is of the form*ax*^{2}+*bx*+*c*= 0, where*a*= 2,*b*= 1, and*c*= -45.**Now, find two real numbers**

*m*and*n*such that,*mn*=*ac*= (2)(-45) = -90 and*m*+*n*= 1.**The two numbers are 10 and -9.**

**Step 2:**

**Sum of the roots = 10 + (-9)**

**= 10 - 9**

**= 1**

**Step 3:**

**Product of the roots = (10)(-9)**

**= -90**

**Step 4:**

**Rewrite the given equation in the form**

*ax*^{2}+*mx*+*nx*+*c*as**2**

*x*^{2}+ 10*x*- 9*x*- 45 = 0**Group the first two terms and the last two terms and then take the common factors out.**

**(2**

*x*^{2}+ 10*x*) - (9*x*+ 45) = 0**2**

*x*(*x*+ 5) - 9 (*x*+ 5) = 0**(**

*x*+ 5)(2*x*- 9) = 0**Step 5:**

**Now, use the zero product property to find the solution.**

**(**

*x*+ 5)(2*x*- 9) = 0

*x*+ 5 = 0 or 2*x*- 9 = 0

*x*= -5 or*x*=^{9}/_{2}**Step 6:**

**So, the solutions are -5 and**

^{9}/_{2}**Practice Questions:**

**Solve the quadratic equations using factoring method.**

**a) x**

^{2}+ 10x – 24 = 0**b) x**

^{2}– 9x – 36 = 0**c) x**

^{2}– 49 = 0**d) 2x**

^{2}– 5x – 12 = 0**e) 6x**

^{2}– 13x + 6 = 0**Answers:**

**a) x = -12 ; x = 2**

**b) x = -3 ; x = 12**

**c) x = -7 ; x = 7**

**d) x = -**

^{3}/_{2}; x = 4**e) x =**

^{2}/_{3}; x =^{3}/_{2}

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