Question

Solve $ {x^2} - 19 $ using square root property.

Answer 1

Hint: In this question we need to solve $ {x^2} - 19 $ using square root property. Here, we will rewrite the given equation, as the numbers are perfect squares. Then, it will be converted into the form of difference of squares. Here, we will apply the formula of difference of squares. And, substitute the value of $ a $ and $ b $ , by which we will get the required answer which will be factors of the given equation.

Complete step-by-step answer:
Now, we need to solve $ {x^2} - 19 $ .
We can rewrite $ 19 $ as $ {\left( {\sqrt {19} } \right)^2} $ .
Therefore, we can rewrite the equation as $ {x^2} - 19 $ ,
 $ {\left( {1x} \right)^2} - {\left( {\sqrt {19} } \right)^2} $
It is known as squaring a square root. Squaring a number and taking the square root of a number are opposite operations; thus, they undo each other. The result of squaring a square root, then, is simply the number under the radical sign.
Now, let us factor using the difference of squares formula,
 $ {a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right) $
Here, $ a = 1x $ and $ b = \sqrt {19} $
Now, by substituting the values, we have,
 $ {\left( x \right)^2} - {\left( {\sqrt {19} } \right)^2} = \left( {x - \sqrt {19} } \right)\left( {x + \sqrt {19} } \right) $
Hence, the factors of $ {x^2} - 19 $ are $ \left( {x - \sqrt {19} } \right) $ and $ \left( {x + \sqrt {19} } \right) $ .
So, the correct answer is “ $ \left( {x - \sqrt {19} } \right) $ and $ \left( {x + \sqrt {19} } \right) $ ”.

Note: In this question it is important to note that a radicand is a number underneath the radical sign. To multiply radicands, multiply the numbers as if they were whole numbers. Make sure to keep the product under one radical sign. Then factor out any perfect squares in the radicand. If it is not a perfect square, then place the square root of the perfect square in front of the radical sign. When the equation is given in the form of a quadratic equation, $ a{x^2} + bx + c $ then we need to find two numbers that multiply to give $ ac $ , and add to give $ b $ which is called sum-product pattern. Then rewrite the middle with those numbers. Then, factor the first two and last two terms separately. If we have done this correctly, then two new terms will have a clearly visible common factor.