Question

How do you factor completely $36{x^2} - 25$ ?

Answer 1

Hint:To order to determine the factors of the above quadratic equation using the identity $\left( {{A^2} - {B^2}} \right) = \left( {A - B} \right)\left( {A + B} \right)$

Complete step by step solution:
Given a quadratic equation $36{x^2} - 25$,let it be $f(x)$
$f(x) = 36{x^2} - 25$
Comparing the equation with the standard Quadratic equation
a becomes 36
b becomes 0
And c becomes -25
To find the quadratic factorization we’ll be writing the expression as
$f\left( x \right) = {(6x)^2} - {(5)^2}$
Consider $6x$as A and $5$as B and Applying Identity $\left( {{A^2} - {B^2}} \right) = \left( {A - B} \right)\left( {A + B} \right)$
Now our equation becomes
$f\left( x \right) = (6x - 5)(6x + 5)$
Hence, We have successfully factorized our quadratic equation.
Therefore, the factors are$(6x - 5)$ and$(6x + 5)$

Alternative:
You can also alternatively use a direct method which uses Quadratic Formula to find both roots of a quadratic equation as
$x1 = \dfrac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}$ and $x2 = \dfrac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}$
x1,x2 are root to quadratic equation $a{x^2} + bx + c$
Hence the factors will be $(x - x1)\,and\,(x - x2)\,$.

Additional Information:
Quadratic Equation: A quadratic equation is an equation which can be represented in the form of $a{x^2} + bx + c$ where $x$ is the unknown variable and a,b,c are the numbers known where $a \ne 0$. If $a = 0$then the equation will become a linear equation and will no longer be quadratic .
The degree of the quadratic equation is of the order 2.
Every Quadratic equation has 2 roots.
Discriminant: $D = {b^2} - 4ac$
Using Discriminant, we can find out the nature of the roots
If D is equal to zero, then both of the roots will be the same and real.
If D is a positive number then, both of the roots are real solutions.
If D is a negative number, then the root are the pair of complex solutions

Note: 1. One must be careful while calculating the answer as calculation error may come.
2.Don’t forget to compare the given quadratic equation with the standard one every time.