**2.3 Quadratic Equations**

1. Quadratic equations are equations which fulfill the following characteristics:

**(a)**Have an

**equal**‘=’ sign

**(b)**Contain only

**one unknown**

**(c) Highest power**of the unknown is 2.

**For example****,**

**2.**The

**general form**of a quadratic equation is written as:

**(a)**

*ax*

^{2}**+**,

*bx*+*c*= 0where

*a*≠ 0,*b*≠ 0 and*c*≠ 0,example: 4

*x*^{2}+ 13*x*– 12 = 0

(b)

(b)

*ax*

^{2}**+**,

*bx*= 0where

*a*≠ 0,*b*≠ 0**but**,*c*= 0example: 7

*x*^{2}+ 9*x*= 0

(c)

(c)

*ax*

^{2}**+**,

*c*= 0where

*a*≠ 0,*c*≠ 0**but**,*b*= 0example: 9

*x*^{2}– 3 = 0**Example 1**:

Write each quadratic equation in the general form.

(a)

*x*^{2}– 5*x*= 12(b) -2 + 5

*x*^{2}– 6*x*= 0(c) 7

*p*^{2}– 3*p*= 4*p*^{2}+ 4*p*– 3(d) (

*x*– 2)(*x*+ 6) = 0(e) 3 – 13

*x*= 4 (*x*^{2}+ 2)(f)
$2-y=\frac{1-3y}{y}$

(g) $\frac{p}{4}=\frac{2{p}^{2}-3}{10}$

(h) $\frac{{y}^{2}+5}{4}=\frac{y-1}{2}$

(i) $\frac{4p}{7}=p(7p-6)$

(g) $\frac{p}{4}=\frac{2{p}^{2}-3}{10}$

(h) $\frac{{y}^{2}+5}{4}=\frac{y-1}{2}$

(i) $\frac{4p}{7}=p(7p-6)$

*Solution:*A quadratic equation in the general form is written as

*ax*^{2}+*bx*+*c*= 0**(a)**

*x*

^{2}– 5

*x*= 12

*x*

^{2}– 5

*x*-12 = 0

**–2 + 5**

(b)

(b)

*x*

^{2}– 6

*x*= 0

5

*x*^{2}– 6*x*–2 = 0**7**

(c)

(c)

*p*

^{2}– 3

*p*= 4

*p*

^{2}+ 4

*p*– 3

7

*p*^{2}– 3*p*– 4*p*^{2}– 4*p*+ 3 = 03

*p*^{2}– 7*p*+ 3 = 0**(**

(d)

(d)

*x*– 2)(

*x*+ 6) = 0

*x*

^{2}+ 6

*x*– 2

*x*– 12 = 0

*x*

^{2}+ 4

*x*– 12 = 0

**3 – 13**

(e)

(e)

*x*= 4 (

*x*

^{2}+ 2)

3 – 13

*x*= 4*x*^{2}+ 8–4

*x*^{2 }– 8 + 3 – 13*x*= 0–4

*x*^{2}– 13*x*– 5 = 04

*x*^{2}+ 13*x*+ 5 = 0**$2-y=\frac{1-3y}{y}$**

(f)

(f)

2

*y*–*y*^{2 }= 1 – 3*y*2

*y*–*y*^{2 }– 1 + 3*y*= 0 –

*y*^{2 }+ 3*y*– 1 = 0*y*

^{2}– 3

*y*+ 1 = 0

**$\frac{p}{4}=\frac{2{p}^{2}-3}{10}$**

(g)

(g)

10

*p*= 8*p*^{2}– 12–8

*p*^{2}+ 10*p*+12 = 08

*p*^{2}– 10*p*– 12 = 0**$\frac{{y}^{2}+5}{4}=\frac{y-1}{2}$**

(h)

(h)

2y

^{2}+ 10 = 4y – 42y

2y

^{2}– 4y + 10 + 4 = 02y

^{2}– 4y + 14 = 0**$\frac{4p}{7}=p(7p-6)$**

(i)

(i)

4

*p*= 7*p*(7*p*– 6)4

*p*= 49*p*– 42^{2}*p*– 49

*p*+ 42^{2}*p*+ 4*p*= 049

*p*– 46^{2}*p*= 0