Question

Factorise the expression \[{x^2} - 64\].

Answer 1

Hint: Factorisation is the process in which an expression is broken up into numbers which in turn can be multiplied to get the original expression. Observe that the above equation is of the form \[{a^2} - {b^2}\], recall the formula and apply it.

Complete step by step solution:
The given quadratic expression is : \[{x^2} - 64\].
To factorize the equation:
Recall the formula \[{a^2} - {b^2}\] \[ = \] \[\left( {a + b} \right)\left( {a - b} \right)\].
Note that the given equation is also of the form \[{a^2} - {b^2}\] \[ = \] \[0\], where \[a = x\] and \[b = 8\], therefore applying the above formula:
\[{x^2} - 64\]
\[ = \] \[{x^2} - {(8)^2}\]

\[ = \] \[\left( {x + 8} \right)\left( {x - 8} \right)\]

Additional information:
Any equation of the form \[a{x^2} + bx + c = 0\], \[a \ne 0\], where \[a,b,c\] are constants and \[x\] is a variable is known as a quadratic equation.
Here, \[{b^2} - 4ac\] is called the discriminant \[D\], of the quadratic equation.
For any quadratic equation:
If \[D > 0\], roots are real and unequal.
If \[D = 0\], roots are real and equal.
If \[D < 0\], roots are imaginary.

Note:
For any quadratic equation we always have two probable solutions for \[x\]. To factorise the above quadratic another method could also be used. For that try to find any two numbers which when multiplied will equal the constant value \[ac\], and when subtracted (in this case, or added if the sign before the constant \[c\] is ‘\[ + \]’) will be equal to \[b\]. Here in this sum \[a = 1,\] \[b = 0\], \[c = 64\], \[ac = 64\] . Factorise \[64\] and try to find the required number. Observe that \[8\] when multiplied by \[8\] equals \[64\]and \[8\] when subtracted from \[8\] gives \[0\] that is equal to \[b\]. Hence both the numbers are \[8\]. Now express \[b\] as the difference of the two numbers as shown:
\[\therefore \] \[{x^2} - 64\] \[ = \] \[{x^2} + 8x - 8x - 64\]
Take out the common factor from the first two terms and last two terms respectively:
\[ = \]\[x\left( {x + 8} \right) - 8\left( {x + 8} \right)\]
Again take out the common factor:
\[ = \] \[\left( {x + 8} \right)\left( {x - 8} \right)\]
Any general quadratic equation can be factorized in this method.