Solving Quadratic Equations

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Malaysia SPM Additional Mathematics, Chapter 2, Quadratic Equation

To solve an equation means to find the root(s) of the equation.
A quadratic equation can be solved by the following methods.

$x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}}$

where a, b and c is the value of the coefficients of the general form $ax^2 + bx + c = 0\,$ .

1 Solving Quadratic Equation by Factorisation
2 Youtube Video

[edit] Solving Quadratic Equation by Factorisation

[edit] When c = 0

The general form of a quadratic equation is $ax^2 + bx + c = 0\,$ . If the constant coefficient, c = 0, the equation can be solved by a simple factorisation. Look at the example below.

Example 1

Solve each of the following quadratic equation by factorisation.

(a) $2x^2 + 3x = 0\,$
(b) $2x = 6x^2\,$

Answer
(a)
$2x^2 + 3x = 0\,$ , factorise the equation
$x(2x + 3) = 0\,$
$x = 0\,$ or $2x + 3 = 0\,$

when
$2x + 3 = 0\,$
$2x = -3\,$
$x = \frac{{ - 3}}{2}$

Hence, the root of the equation is x = 0 or x = -3/2.

(b)
$2x = 6x^2\,$
$2x - 6x^2 = 0\,$ , factorise the equation
$2x(1 - 3x) = 0\,$
$2x = 0\,$ or $1 - 3x = 0\,$
when
$1 - 3x = 0\,$
$1 = 3x\,$
$x = {1 \over 3}$

Hence, the root of the equation is x = 0 or x = 1/3.

[edit] When b = 0

Example 2

Solve the following equation.

(a) $9x^2-36=0\,$
(b) $98 - 2x^2 = 0\,$
(c) $3x^2 - 6 = 0\,$
Answer
(a)
$9x^2-36=0\,$
$(3x)^2-6^2=0\,$
$(3x + 6)(3x - 6)=0\,$
$3x + 6 = 0\,$ or $3x - 6 = 0\,$
$3x = -6\,$ or $3x = 6\,$
$x = -2\,$ or $x = 2\,$

(b)
$98 - 2x^2 = 0\,$
$49 - x^2 = 0\,$
$7^2 - x^2 = 0\,$
$(7 - x)(7 + x) = 0\,$
$7 - x = 0{\rm \quad or\quad }7 + x = 0\,$
$x = 7{\rm \quad or\quad }x = - 7\,$

(c)
$3x^2 - 6 = 0\,$
$3x^2 = 6\,$
$x^2 = {6 \over 3}$
$x^2 = 2\,$
$x = \pm \sqrt 2$
$x = \sqrt 2 = 1.414$
or
$x = -\sqrt 2 = - 1.414$

[edit] When b ≠ 0 and c ≠ 0

Example 3

Find the roots of the following quadratic equation.

(a) $x^2 + x = 20\,$

(b) $3x^2 - 22x + 7 = 0\,$

(d) $( 2x + 1)( x + 6 ) = 20x\,$

Answer
(You should have learned how to factorise q quadratic Expression in form 3. You if have forgotten, you may visit the MyHomeTuition.com's online tuition webpage to revise. You may also click on the Youtube video below to learn solving quadratic equation by factorisation)
(a)

$x^2 + x = 20\,$

$x^2 + x - 20 = 0\,$

$(x - 4)(x + 5) = 0\,$

Hence,

$(x - 4)= 0\,$ or $(x + 5) = 0\,$

$x = 4\,$ or $x = -5\,$

$x = 1/3\,$

(b)

$3x^2 - 22x + 7 = 0\,$

$(3x - 1)(x - 7) = 0\,$

Hence,

$(3x - 1)= 0\,$ or $(x - 7) = 0\,$

$3x = 1\,$ or $x = 7\,$

$x = 1/3\,$

The equation is not in general form. To factorise a quadratic equation, we must put the equation in general form.
There are brackets in the equation. We must expand the brackets before we can rewrite the equation in general form.

$( x + 1 )( x - 2 ) =4x - 8\,$