Reviewing quadratic equations
Clement Radcliffe,Contributor

I do hope that you will have anenjoyable christmas holiday. It would be more beneficial if you are able to find the time to review your mathematics lessons. Remember the examination is about FIVE MONTHS away.

We will continue with the review of ALGEBRA. Let us solve together the following quadratic equations.

Solve the following:

• x² - 9x + 14 = 0. Factorising the left side

x² - 9x + 14 = (x - 2)(x - 7)

... (x - 2)(x - 7) = 0

... x - 2 = 0, that is, x = 2 OR x - 7 = 0, that is, x = 7

Answer: x = 2 and 7

• 2x² - x - 15 = 0

... (2x + 5)(x - 3) = 0

... 2x + 5 = 0, that is, x = 5/2 OR x - 3 = 0, that is, x = 3

Answer: x = -5/2 and 3

• x² + x = 6

... x² + x - 6 = 0

... (x + 3)(x - 2) = 0

... x = -3 and 2

• Solve:y = 2x² - 3x - 2 when y = 0

... y = 2x² - 3x - 2 when y = 0

... y = 2x2 - 3x - 2 = 0

Factorising

(2x + 1)(x - 2)=0

... x = -1/2 and 2

Most quadratic equations cannot be solved by factorisation. Alternatively, the FORMULA METHOD is used. Please be reminded that given the quadratic equation ax² + bx + c = 0, where a, b and c are constants, then it can be shown that x =
( (-b (+ or -) square root of b2 - 4ac)/2a).

This is the basis of the formula methid as x is found by substituting the values of a, b and c into the formula.

Examples:

• Solve x² - 3x - 1 = 0 correct to two decimal places. As the quadratic equation is expressed in the form ax² + bx + c = 0 then a = 1, b = -3 and c = -1. Substituting into the formula

x = ( (-b (+ or -) square root of b2 - 4ac)/2a) then x = ( (-3 (+ or -) square root of -3² - 4 x 1 x -1)/2 x 1)

... x = ( (3 (+ or -) square root of 9 + 4)/2 )

... x = (3 + 3.61)/2 = 6.61/2 = 3.31

x = (3 - 3.61)/2 = -0.61/2 = -0.31

Answer x = 3.31 and -0.31

• Express 2x² = 3x + 1 in the form ax² + bx + c = 0 and find the values of a, b and c.

Given that 2x² = 3x + 1, then 2x² - 3x - 1 = 0.

By comparing this equation with the required form ax² + bx + c = 0

... a = 2, b = -3 and c = -1

Please be careful not to omit the negative sign.
Answer: a = 2, b = -3 and c = -1.

• Solve 2x² - 3x - 7 = 0. Using the Formula method:

From the equation, a =2, b = -3 and c = -7.

(Note that the zero must be on the right hand side).

Given the formula: x = ( (-b (+ or -) square root of b2 - 4ac)/2a) , then substituting

... x = ( (-(-3) (+ or -) square root of -3² - 4 x2 x (-7))/2 x2)

... x = ( (3 (+ or -) square root of 9 + 56)/4 )

= ( 3 (+ or -) square root of 65) / 4 = ( 3 (+ or -) + 8.063) / 4

... Either x = 11.063/4 OR x = -5.063/4

... x = 2.766 OR -1.266

Let us try another example.

Solve the following equation using the quadratic formula: 2x² + 2x - 8 = 3x - 6

2x² + 2x - 8 = 3x - 6
2x² + 2x - 3x - 8 + 6 = 0
2x² - x - 2 = 0

Having expressed the equation into the appropriate form, then a = 2, b = -1 and c = -2.

Using the formula: x = ( (-b (+ or -) square root of b2 - 4ac)/2a)

... x = ( (-(-1) (+ or -) square root of (-1)² - 4 x2 x -2/2 x2)

... x = ( 1 (+ or -) square root of (-4 x 2 x -2/4) = ( 1 (+ or -) square root of 1 + 16/4)

... x = ( 1 (+ or -) square root of 17/4) = ( 1 (+ or -) 4.12/4)

... x = ( 1 + 4.12)/4) = 5.2/4 = 1.28

And x = ( 1 - 4.12)/4) = - 3.12/4 = 0.78

Answer is x = 1.28 and -0.78

Unless you are specifically directed, you should attempt to use the factorisation method before the Formula method. If you are asked to give the solution of a quadratic equation correct to two decimal places, then this is an indication that the formula method be used.

POINTS TO NOTE

Care should always be taken in manipulating the negative signs, as this provides the greatest challenge in this method.

The (+ or -) enables you to obtain two roots

The entire numerator is over 2a. A common error is to use (square root of b² - 4ac) over 2a, seperating -b. In other words, the incorrect formula ( -b (+ or -) b² - 4ac) / 2a

The value within the the square root should always be positive. When this is not so, it usually implies an error in calculation. PLEASE CHECK YOUR WORKING.

If the value within the square root is negative, then the equation has no real roots.

For homework, please find the solution of the quadratic equations.

(1) x² + 3x + 1 = 0

(2) 2x² - 6x - 1 = 0

(3) 7x² + 8x - 2 = 10

(4) 2x² - 3x - 4 = 2 - 4x

The Young Men's Christian Association's tae kwon do group run laps at the 30th annual Heart Health Fund Run, held on November 29 at the National Stadium east field. - Contributed

Clement Radcliffe is principal of Glenmuir High School. Send questions and comments to kerry-ann.hepburn@gleanerjm.com