Solve. √(4 c-7)=√(4 c+1)-4

Name: Problem 60, Chapter 10: Beginning and Intermediate Algebra
Uploaded: 2021-04-30T02:58:02-08:00
Duration: 3 min 1 s
Channel: Nick Johnson
Description: Solve. √(4 c-7)=√(4 c+1)-4

Key Concepts

Radical Equations

A radical equation contains variables inside a radical, typically a square root. Solving such equations often involves isolating the radical term and then eliminating the radical by raising both sides of the equation to a power, usually squaring, to transform the equation into a more familiar algebraic form.

Domain Restrictions

When solving radical equations, it is essential to consider the domain of the radical expression. Since the square root function is defined only for non-negative arguments, the expressions under the radical must be greater than or equal to zero, which imposes necessary conditions on the values of the variable.

Extraneous Solutions

Squaring both sides of an equation can introduce solutions that do not satisfy the original equation. These false or extraneous solutions must be checked against the original equation after solving to ensure that only valid solutions are accepted.

Isolation and Squaring Method

A common strategy for solving radical equations is to first isolate the radical term and then square both sides to remove the radical. Once the equation is free of radicals, it simplifies into a polynomial equation that can be solved using standard algebraic techniques. After obtaining the solutions, it is important to substitute them back into the original equation to verify their validity.

Transcript

00:01 Okay, so we have the radical equation.

00:04 The square root of 4c minus 7 is equal to the square root of 4c plus 1 minus 4.

00:10 So we first go ahead and square both sides.

00:13 So the left -hand side, squaring it just gives us what's underneath the radical or the radic hand.

00:18 So that just gives us 4c minus 7.

00:21 Okay, but then squaring the right -hand side, that's going to give us, well, well, 4c plus 1, but then we get a minus 8 times the square root of 4c plus 1, and then a plus 16, right? that's because we have a squared plus b squared, that's equal to a squared, and then minus 2 times ab plus b squared.

00:50 So that gives us this here.

00:52 Okay, and then we combine our like terms and simplify, and we end up with negative 24 is equal to negative 8 times the square root of 4c plus 1.

01:10 Okay, now we can go ahead and square both sides to then eliminate the radical.

01:16 So we'd end up with negative 24 squared.

01:19 That's going to be 576, and that's going to be equal to, well, 64 times just was under the radical, so 4c plus 1.

01:33 The 64 then distributes, so we end up with 576 is equal to, well, 64 times 4c.

01:42 That gives us a 256c, and then 64 times 1, so plus 64 times 1, so plus 64.

01:50 Okay, so then combine our like terms here, subtract 64 from both sides, and we get 512, or, yeah, 512, 576 minus 64, so 512, is equal to 256 c...

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