Solve each equation. (8-√(a))^2+6(8-√(a))-7=0

Name: Problem 71, Chapter 10: Elementary and Intermediate Algebra
Uploaded: 2020-04-19T12:42:42-08:00
Duration: 5 min 10 s
Channel: Shaza Hammoud
Description: Solve each equation. (8-√(a))^2+6(8-√(a))-7=0

Key Concepts

Radical Equations and Domain Considerations

When dealing with equations involving square roots or other radicals, it is important to consider the domain of the expression under the radical to ensure that the solutions are valid. Since the square root function is only defined for non-negative arguments in the real number system, any substitution involving a square root must respect that a must be non-negative. Analyzing the domain helps avoid extraneous solutions that might arise during the manipulation of the equation.

Quadratic Equations

This concept involves solving equations where the unknown appears in a squared term along with linear and constant terms. The general form is ax² + bx + c = 0, and such equations can be solved by factoring, completing the square, or using the quadratic formula. Recognizing a quadratic structure allows one to apply these standard methods to find the roots, even when the variable has been substituted or transformed.

Substitution Method

Substitution is a powerful algebraic technique used to simplify equations by replacing a complicated expression with a single variable. In problems where a composite expression is repeated, substituting it with a new variable can reduce the equation to a familiar form, such as a quadratic equation. After solving the simplified equation, one substitutes back to recover the solutions in terms of the original variable.

Transcript

00:01 And this question asks to solve the following equation.

00:04 As you can see, the leading term and the middle term has an expression, which is 8 minus squared of a.

00:11 So let's simplify this equation by using proper substitution.

00:18 I'll substitute instead of this whole expression, 8 minus squared of a, i'll put one single variable.

00:27 So i'll set that x will be equal to.

00:30 To 8 minus square root of a.

00:35 And this it means that instead of each 8 minus squared of a in this expression, it will become x.

00:45 So x, 4 of 2 is the leading term, plus 6.

00:49 Also here will become x minus 7 equals to 0.

00:54 So i got an equation with one unknown x, and this is a question.

01:02 Equation so let's solve it using one of the metals i'll use here the the the factoring method i want two numbers to be multiplied by each other and to get in order to get minus seven and by adding these two numbers i'll get positive six so what are the factors of minus seven they are just one and seven so to get minus seven it means that one of them should be negative so which one is negative, okay, we can determine that by looking at the sign of the coefficient of the middle term.

01:41 So it's plus six.

01:43 How to get plus six from one and seven? it can be done by that one is negative and seven is positive as minus one plus seven will get to positive six.

01:55 So minus one and seven is positive as if it's at the opposite and one is positive, seven is negative.

02:04 Seven is negative.

02:04 Then it's 1 minus 7 and this will lead to negative 6 not positive 6 and this is incorrect now this so these are two factors minus 1 and positive 7 so let's write them in the form of x minus 1 x plus 7 equals to 0 by sitting each factor to be equals to 0 x minus 1 equals to 0 this leads to x equals 1...

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