![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbtowQ3CzlwRJZqbG_YP2mhZalGl_dIeuKbPW-uBNNUcpO6DAyOVRXnphKDuOEs6WLUj1AacMWZJxEioVHQkEDccwH6M10yi5aJFarZLi3VSpdC3vSROs9dK-CMlqna3INY1XoIrH5nSLB/s280/c4743r+%28Blackboard%29.jpg)
Picture is obtained from http://foldspace.files.wordpress.com/2008/06/c4743r.jpg
The mathematics question is,
Find the value of k of the quadratic equation
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXO-zkTKB9ZRa4hiv43-Mm_DsIrD0zdHHruCstlyUeZFmO3DfX6lwdgKO6f2NohRVBdHtxMSWN_cVdhvIXMjb7WtAmxAV7zPrctjfX6-POUYroyx5pEzAjUoxh4dWRD3UJqx5RpxVUDCP9/s400/Quadratic+Equation+-+Question.jpg)
Such that one of its roots is thrice the value of the other.
Solution:
Let α and β be the roots. We have,
Sum of roots:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEid5h-1wSEibYYdCLzyGYn_0t6aVp69sU0ILZJZ3Ho-BSk9Hr_9RPhYKRzK1qU3-YkEXWxODOiAGPy_AKxUpAIXUCMpZ42A3E5DC6Mp3i4KHwpsd4fG7rtHGyuyAAq1ZlwGcqXqYlspgjek/s280/Sum+of+Roots.jpg)
Product of roots:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwwahZwvc8vmQwDlnDGryunFgAIunuLcK6krKfC1mHMKVFUZ3DKIqOZseHsPlDmUryWrvSoDSAd_P0z9RqclLicbPyf6E17GIfKE9MWFEo81w9Nv-qIBha_PI3z8P-CxFdSYVs-SyeDX_4/s280/Product+of+Roots.jpg)
Given that,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfhtvRMaIZDQY5D9ndSUFXrC40SBbG26TvEhstPJdAANoRnVgA4GScyaqiWrZdRlchxJjqntnZee1fDTeGN28AAp_yPXizuLx13A9jZsED5zkyONh65J-dTG1jTsUAJg3JUYqXTTELTaFp/s280/One+root+is+thrice+the+value+of+the+other.jpg)
Substituting into eq. (1), we get
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2zdygYGLZiSiNxxOfgaDWUH94o-IxHa5URLprW73fH8KGc8Ok_gAd431jFDXJAf4p5CwxnSFszX29P6_3i9T7uSLxriBZnuEL4WFkU7Bgs9RcnUJEt4U1JVCwM2ZHWondnNXkJwSMtar7/s400/Substituting+into+eq+%281%29.jpg)
Therefore the other root is,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFlUuiJ-xJATj40A-l0JhVhfURXhTH34ChREqAw_kMx2BAODahVFvugFWTqH-tnQ_ADSSkCwRdMZateOQEzpX4wphRgABePugrSTybQhB4B4jjS6eq2u7fOLZG-U41famY75REBnXwlkoF/s400/Value+of+the+other+root.jpg)
What to do next?
The roots, α and β are found but in terms of k.
How to find the value of k?
Note that equation (2) is not yet used.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzCo5IQfNFdG-ZTmPON76lem4xDgofLMlwSBomZnNITqLOOR47bE0vmyarquxsoy0Ljaqc2g6n0Zd2Cd9tTnNF7xbuiaZ1ERH8VmgVJtg6yaE_A88jLUIgV-0jyiQmetsf1FJayq6jnK2B/s400/Quadratic+%28Scary%29.jpg)
Picture is obtained from http://plus.maths.org/issue29/features/quadratic/Quadratic.jpg
Substituting α and β into eq. (2), we have,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnkIREdStSUkL3JOKdJR3J5oXYT7JhEN4GJ8wZZovmBpDX0v-A6V-owdc5R7XVT05rjDjHnCv0dMdtlni9Fmr0YqyrzAdBTfIN53scanH07j7-gX-GGbpIMCVN5J4zrGhlevW8PibofWnT/s400/Substituting+the+roots+into+eq+%282%29.jpg)
How to obtain the Quadratic Formula
From Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Quadratic_equation)
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjugZkWlK7L7DQgNW53LZFAw-6DMAZmKFby8V8fCfogSajnjgsDmAQwGXyw1r-7u6dVLaVGeH0SKHsgiuSBHmU2v41IHyNQdc9fkxgSh6rtKxL55gClmVPla89SprcIAlBHM6dQL3B2b9zM/s400/2nd+degree+polynomial+equation.jpg)
The quadratic coefficients are the following:
a is the coefficient of x2
b is the coefficient of x
c is the constant coefficient (also called the free term or constant term).
Quadratic equations are called quadratic because quadratus is Latin for "square".
The roots x or the quadratic formula can be derived by the method of completing the square. Dividing the quadratic equation
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjugZkWlK7L7DQgNW53LZFAw-6DMAZmKFby8V8fCfogSajnjgsDmAQwGXyw1r-7u6dVLaVGeH0SKHsgiuSBHmU2v41IHyNQdc9fkxgSh6rtKxL55gClmVPla89SprcIAlBHM6dQL3B2b9zM/s400/2nd+degree+polynomial+equation.jpg)
by a (which is allowed because a is non-zero), gives:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgICTO0wmin1ZujM0r7-4fHiRHcXO4W2lYjfOsGwRsmB-T88mXskyCI0k2_awCZZIW7GI39sWfUUgXw2gJqG7KolbDHd3Ub1THDktnIMSSoXaJRHdaYiSfkP8tseeUvuiCl32Iwnn5UAgiV/s400/2nd+degree+polynomial+equation+divided+by+a.jpg)
or
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8XeVQco93_D1CRem4T3Cp1FTdkRWzvjYoWwQenwPYLwDhEcCqNevnZlOIMhcOZygvX47r06rbUNAnJ4oa5Jv8oFsWeXDUEw6WTCXgbZ3Jr7LGeLQNfcvOWieSRsXL_qQdEeBs9MgXGZCM/s400/2nd+degree+polynomial+equation+divided+by+a+rearranged.jpg)
The quadratic equation is now in a form to which the method of completing the square can be applied.
To "complete the square" add the constant,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdqciBYfD8DIGkRDqtTe0uW-K7HeA2WTupQwg91pNtwrPnaXiTJpklQSZrx2NbbaEwa9SJ-vAjoPM5aAcOU06lQ2iOl__LHNf4ICDZeLK7kIC7RJNJ1qkWVCJQUSD5G3cEQq3Ho3Vah0Yd/s400/Constant+for+Completing+the+Squares.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjKX4q8CVRjLj5X5mo4V7mKyibgQDh3WFjbsaURX840jMQ3YJnKDsvkqLH87ePzfLynBU7-m4jdll8kTMF5x7NW0uDKYwbllWyhyphenhyphenAF0W8fK7PhapgisrvIdgXeOB_6UkSIYxArP4KOFX8F/s400/2nd+degree+polynomial+equation+with+constant+added+to+both+sides.jpg)
The left side is now a perfect square because
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8L-WYIWzV7kV5hmSIZHbgt13NBOwXTuGIr08x_QziQ1_cGB1_EFN4aY_6J2CMEKeo_qdGYDaw2CVaZsht5xUU7ec9wFsNM6sCpNU808Z3mCfzRKQfGKj2PEdcWaRxGnprm7RCXgcKP-FJ/s400/LHS+is+a+perfect+square.jpg)
Substituting the left side with the perfect square we have,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHs6_ZOkg_hzq6raEhe4AU6kFEWZJvI9cBxW7iS0L6uyZzcV4LvpMUDr3VBoY-x8AJYBeteLqomOyyKPBGIn-ogP6dGP_PRYFeoH43hCyrrTH17OysnlQkpdJf-F5JY8BLCzlACdCWsdsw/s400/Completing+the+Squares+-+Perfect+Squares+at+LHS.jpg)
The right side can be written as a single fraction, with common denominator 4a2. This gives
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif9cs8ex7v5twI68S56AL63pUK15nVh0zTHn-Kx7VQCuKnBqZhG5CkG48LkouorpNE4AEsimI73EWnSZH3_nAeEy-B5RtpLnM_aZYgOviQKOXimpsYS0Xv_HJEWTuCi49GQCQs2pyKm3Fr/s400/RHS+expressed+as+fraction.jpg)
Taking the square root of both sides yields
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijHsLPueXtlXrlvwKNrl0Fb_XjLhkSQfUr_qwLGsuFKbrvXHL06E0HTv1Oy13ClH5a4SHRkgxv5ovXE3yLNM3Mio2gqx0Y11JeUC24E8GJtN9g6JfHcbNxvhRXbM4v6g7YV3pldp6H4OCP/s280/Taking+square+root+for+both+sides.jpg)
Isolating x, gives
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQHu1VQhZj5ieqp6VohzFoBQlltQ7UlP5lE26mxKdfDNeypx9W88ViVO0Olks-bdKVSRntL6bzjSamUR-p3rqAFeBWvQqJcfUIQnrNLlNj3JDc6miipIYW14kisiYt6EdD-xDdAvqQbZZA/s400/Isolating+X,+to+obtain+the+Quadratic+Formula.jpg)
This equation is known as the quadratic formula.
How to obtain Sum of roots and Product of roots
The general form is of a quadratic equation is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjl8t7Lk3wMqDwxg4PsEq1TdKp2q_ncYSjw3rPqQOo-Jaqlx_450gXwK6oRmsuUaypMsu8EFzyQ-LidVfEf8q1_Tfwc6TIc1OBPg6O0ALoIEI4goJrJ3_1VNM9FagmjnsGSIskKl31HVTxj/s400/Sum+%26+Product+of+roots+-+General+Quadratic+Equation.jpg)
Dividing the quadratic equation by coefficient, a, we have
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_lLX08hbQOUwMVb_1s1iT2qhi3rkKVO18j0jKRFyLAK7zApwg5uNLCB99lRb9SaYJbjY72MwL-yMlT0MIyXA_4D6-YpLc2gjhhD0m0LrzAOs8zmQ8ZVrfhunfzIFebvtp_5b0K5SZlUCZ/s400/Sum+%26+Product+of+roots+-+Equation+divided+by+a.jpg)
If a quadratic equation has roots, α and β, the equation can be written as
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiA9OCiAonGkDlC0ts7E6CdZwvMSMOREVIU_tDB4Y2MWZ8rYnJ6859vXq3vLUEIelNnj2rW9JSQs6B0TrGSfLjBooH2QlaTnBsIm0OmmLOPyi_8s497k0kSdhedoBuMn8Vc1nqVSeHbA3ph/s400/Sum+%26+Product+of+roots+-+Equation+with+%CE%B1+and+%CE%B2.jpg)
That is,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyFLZMqIgJIVCH6KbLSmNw3N46pBQME2H3KuWJHwbfagCqYvJ8keyp_v5v9Yx5nd7yiNlBgnRzJG5Ts98ARIlU6XDKajo9TkijvW6nVfye_s5fKreWA9LjQgCgvcL6ijag_pcq0LAwGw7T/s400/Sum+%26+Product+of+roots+-+Equation+with+%CE%B1+and+%CE%B2+expanded.jpg)
Comparing eq.(1) and eq.(2),
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-9awiKb4M7MxknvD87c2-ZCQe14qtJ8AubrIWKRpCYJnFsg8t3Rg3uz_q2ybqypABqZOszBhpX_y6zzlKoGS-kubB_xs09o5nW2w8AZm7kNRQAD6B-dpG0gXwzEjcp3dq3-94f6oqVEao/s400/Sum+%26+Product+of+roots+-+Final+Results.jpg)
Reference
- http://en.wikipedia.org/wiki/Quadratic_equation
- Additional Mathematics, 8th Edition, by The Keng Seng, Loh Cheng Yee, Consultant: Dr. Yeap Ban Har, Shinglee Publishers Pte Ltd, Chapter 2 - “Quadratic Equations and Functions”, Page 32
- http://foldspace.files.wordpress.com/2008/06/c4743r.jpg (Blackboard)
- http://plus.maths.org/issue29/features/quadratic/Quadratic.jpg (Scary)