An expression of the typea x²+b x+ c= 0 , ( a ≠ 0 )is called a quadratic equation in the variablex .
The equation a x ² + b x + c = 0 is called the general (or, standard form) We can solve a quadratic equation by (1) factorization or by (2) applying the formula.
The formula of finding the roots of the quadratic equation is as follows x= (- b ± √ (b ² - 4 a c) ) / 2 a Now we will discuss how to solve applied problems.
Due to wide variety of applied problems, there is no single solving technique that works in all cases.
However the following suggestion proved helpful.
Step: 1Read the problem carefully and determine what quantity (s) must be found.
Step: 2assign a variable name to the quantity.
Step: 3try expressing the problem algebraically, and as well determining which expressions are equal and write the necessary equation (s).
Step: 3solve the resulting equation (s) Now go through a simple problem based on formation of quadratic equation and solving Problem: The denominator of a fraction is one more than twice the numerator.
If the sum of the fraction and its reciprocal is 58 / 21.
Find the fraction.
Solution: Let the numerator of the fraction be x (x be an element of I ) Then its denominator is (2x + 1).
So the fraction isx/ (2x + 1)) And the reciprocal would be (2 x + 1) / x According to the problem: (X / (2 xs + 1)) +((2 x + 1) / x) = ( 58 / 21 ) (X ² +(2 x + 1) ²) /(x + (2x + 1))= ( 58 / 21) [L.
C.
Dis =( x +(2x+ 1 ) ] 21 ( x² +4 x²+ 4 x+1)=58x ( 2x + 1 ) 105 x² + 84 x + 21 =116 x ² + 58 x 11 x ² - 26 x - 21 = 0 11 x ²- (33 - 7) x - 21 = 0[using middle term factorization] 11 x ²- 33 x + 7 x - 21 =0 11x(x-3) +7(x-3) = 0 (x - 3)(11 x + 7) = 0 Either ( x -3) = 0,or (11x+7 ) =0[ usingzero factor theorem ] x = 3, From, (11x + 7) = 0 We get,x =- (7 / 11) But x is an integer , neglect x = - (7 / 11) Take x = 3 So the required fraction, (x / (2x + 1)) = (3 / (2 * 3+ 1)) =(3 / 7) Now try the following: The age of a man is twice the square of the age of his son .
Eight years hence the age of the man will be 4 years more than thrice the age of his son.
Find their present age? If you cannot solve this problem, you probably need more practice.
A good online tutor would be helpful if you plan to master this subject in a short period.
Any reasonably good math tutor should do.
The equation a x ² + b x + c = 0 is called the general (or, standard form) We can solve a quadratic equation by (1) factorization or by (2) applying the formula.
The formula of finding the roots of the quadratic equation is as follows x= (- b ± √ (b ² - 4 a c) ) / 2 a Now we will discuss how to solve applied problems.
Due to wide variety of applied problems, there is no single solving technique that works in all cases.
However the following suggestion proved helpful.
Step: 1Read the problem carefully and determine what quantity (s) must be found.
Step: 2assign a variable name to the quantity.
Step: 3try expressing the problem algebraically, and as well determining which expressions are equal and write the necessary equation (s).
Step: 3solve the resulting equation (s) Now go through a simple problem based on formation of quadratic equation and solving Problem: The denominator of a fraction is one more than twice the numerator.
If the sum of the fraction and its reciprocal is 58 / 21.
Find the fraction.
Solution: Let the numerator of the fraction be x (x be an element of I ) Then its denominator is (2x + 1).
So the fraction isx/ (2x + 1)) And the reciprocal would be (2 x + 1) / x According to the problem: (X / (2 xs + 1)) +((2 x + 1) / x) = ( 58 / 21 ) (X ² +(2 x + 1) ²) /(x + (2x + 1))= ( 58 / 21) [L.
C.
Dis =( x +(2x+ 1 ) ] 21 ( x² +4 x²+ 4 x+1)=58x ( 2x + 1 ) 105 x² + 84 x + 21 =116 x ² + 58 x 11 x ² - 26 x - 21 = 0 11 x ²- (33 - 7) x - 21 = 0[using middle term factorization] 11 x ²- 33 x + 7 x - 21 =0 11x(x-3) +7(x-3) = 0 (x - 3)(11 x + 7) = 0 Either ( x -3) = 0,or (11x+7 ) =0[ usingzero factor theorem ] x = 3, From, (11x + 7) = 0 We get,x =- (7 / 11) But x is an integer , neglect x = - (7 / 11) Take x = 3 So the required fraction, (x / (2x + 1)) = (3 / (2 * 3+ 1)) =(3 / 7) Now try the following: The age of a man is twice the square of the age of his son .
Eight years hence the age of the man will be 4 years more than thrice the age of his son.
Find their present age? If you cannot solve this problem, you probably need more practice.
A good online tutor would be helpful if you plan to master this subject in a short period.
Any reasonably good math tutor should do.