Algebra

Quadratic Equations

If 𝑎𝑥²+𝑏𝑥+𝑐=0, 𝑥=−𝑏±𝑏²−4𝑎𝑐2𝑎 If 𝑎𝑥²+2𝑏𝑥+𝑐=0, that is, if the coefficient of 𝑥 be an even number, 𝑥=−𝑏±𝑏²−𝑎𝑐𝑎

Method of solution without the formula

Ex. 2𝑥²+7𝑥+3=0 Divide by 2, 𝑥²+72𝑥+32=0 Complete the square, 𝑥²+72𝑥+74²=74²−32=2516 Take square root, 𝑥−74=±54 𝑥=7±54=3 or 12 Rule for completing the square of an expression like 𝑥²+72𝑥, add the square of half the coefficient of 𝑥.
The solution of the foregoing equation, employing the formula is 𝑥=−𝑏±𝑏²−4𝑎𝑐2𝑎=−7±49−244=7±54=3 or 12

Theory of Quadratic Expressions

If 𝛼, 𝛽 be the roots of the equation 𝑎𝑥²+2𝑏𝑥+𝑐=0, then 𝑎𝑥²+2𝑏𝑥+𝑐=𝑎(𝑥−𝛼)(𝑥−𝛽) Sum of roots 𝛼+𝛽=−𝑏𝑎 Product of roots 𝛼𝛽=𝑐𝑎 Condition for the existence of equal roots: 𝑏²−4𝑐𝑎 must vanish.

Examples

The solution of equations in one unknown quantity may sometimes be simplified by changing the quantity sought. 2𝑥+3𝑥−13𝑥+1+18𝑥+66𝑥²+5𝑥−1=14 6𝑥²+5𝑥−13𝑥+1+6(3𝑥+1)6𝑥²+5𝑥−1=14 Put 𝑦=6𝑥²+5𝑥−13𝑥+1 Thus 𝑦+6𝑦=14 𝑦²−14𝑦+6=0 𝑦 having been determined from this quadratic, 𝑥 is afterwards found from derived equation.

Examples

𝑥²+1𝑥²+𝑥+1𝑥=4 𝑥+1𝑥²+𝑥+1𝑥=6 Put 𝑦=𝑥+1𝑥

Examples

𝑥²+𝑥+322𝑥²+𝑥+2=𝑥2+1 2𝑥²+𝑥+3 2𝑥²+𝑥+2=2 2𝑥²+𝑥+2+3 2𝑥²+𝑥+2=4 Put 2𝑥²+𝑥+2=𝑦, and solve the quadatic 𝑦²+3𝑦=4

Examples

³𝑥^𝑛+23³𝑥^𝑛=163𝑥^−𝑛 𝑥^4𝑛3+23𝑥^2𝑛3=163 A quadratic in 𝑦=𝑥^2𝑛3

Find Maxima and Minima Values

Given 𝑦=3𝑥²+6𝑥+7, to find what value of x will make 𝑦 a maximum or minimum. Solve the quadratic equation 3𝑥²+6𝑥+7−𝑦=0 Thus, 𝑥=−3±3𝑦−123 In order that 𝑥 may be a real quantity, we must have 3𝑦 not less than 12; therefore 4 is a minimum value of 𝑦, and the value of 𝑥 which makes 𝑦 a minimum is -1.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive