Solve. √(4 p+5)+√(p+5)=3

Name: Problem 70, Chapter 3: Algebra and Trigonometry
Uploaded: 2020-07-13T09:30:52-08:00
Duration: 7 min 2 s
Channel: Ankit Gupta
Description: Solve. √(4 p+5)+√(p+5)=3

Key Concepts

Radical Equations

Radical equations involve variables under the square root or other radical signs. Solving these equations typically requires isolating the radical expression and then raising both sides of the equation to the appropriate power to eliminate the radical, with careful attention paid to preserving the equality and considering potential restrictions.

Domain Considerations in Square Roots

When working with square root functions, it is important to ensure that the expressions under the radical are non-negative, as square roots of negative numbers are not real. This means that constraints must be applied to the variable so that the radicands (the expressions inside the square roots) are greater than or equal to zero.

Squaring and Extraneous Solutions

Squaring both sides of an equation is a common method used to remove square roots. However, this process can introduce extraneous solutions—solutions that satisfy the squared equation but not the original equation. Therefore, after squaring, it is essential to substitute the solutions back into the original equation to verify their validity.

Transcript

00:01 Our question is square root of 4p plus 5 plus square root of p plus 5 is equal to 3.

00:13 We have to solve the equation and find out the value of p.

00:16 The first step we will do is we will isolate one radical.

00:20 We will get square root of 4p plus 5 is equal to 3 minus square root of p plus 5.

00:26 Now, using the principle of powers, using the principle powers, we will square both the sides.

00:43 We will get 4p plus 5 is equal to 9 minus 6 square root p plus 5 plus p plus 5.

00:54 We will get 4p plus 5 minus 9 minus p minus 5 is equal to minus 6 square root of p plus 5.

01:05 We will subtract 9 p and 5 from both the side.

01:10 So now combining the like terms we will get 3p minus 9 is equal to minus 6 square root p plus 5.

01:19 Now again using the principle of powers we will square both the sides.

01:24 We will get 9p square minus 54 p plus 81 is equal to the square of minus 6 is 36 in bracket p plus 5 9p square minus 54 p plus 81 is equal to so 36 will be multiplied to p 36 p plus 36 plus 36 multiplied by 5 will give you 180 now we will subtract 36p and 180 from both the sides.

01:59 So we will get 9p square minus 54p plus 81 minus 36p minus 180 is equal to 0.

02:14 So combining the light terms we'll get 9p square minus 90 p minus 54 p.

02:24 And minus 36 p will give you 90p plus 81 and minus 180 will give you minus 99 is equal to 0 so we have got the quadratic equation we will factor it to find out the value of p so first we'll see that 9 is the greatest common factor so 9 in bracket p square minus 10 p minus 11 is equal to 0 now we'll factor it 9 then again p square minus 11 p plus p minus 11 will give you 0 9 in bracket p p minus 11 plus 1 p minus 11 then bracket close is equal to 0 now what we will get 9 will be written as it is then p plus 1 and p minus 11 is equal to 0 now we will set each factor equal to 0 and find out the value of p so 9 is equal to p this is wrong because this is contradiction statement as 9 can never be equal to 0 next p plus 1 is equal to 0 will give you p is equal to minus 1 p minus 1 p minus 11 is equal to 0 will give you p is equal to 11.

03:49 So we have got two values of p minus 1 and 11.

03:53 Now we will see which values satisfy our equation.

03:57 So for that we will check...

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