Use backtracking to solve the equations in Parts a and b. a. (3 x+4)^2=25 b. (3 x+4)^2=0 c. How

step 1 For this question, it's required to solve the following equation. However, it's clear that this

Use backtracking to solve the equations in Parts a and b. a....

Key Concepts

Square Root Property

When an equation involves a squared expression, the square root property is a key tool. Taking the square root of both sides of an equation reverses the squaring operation, but it also introduces the possibility of two solutions, one positive and one negative, because both a number and its negative have the same square. This is why careful handling of plus and minus signs is essential when solving equations of the form (expression)² = constant.

Handling Zero in Squared Equations

Equations where a squared term equals zero represent a special case. Since the square of any real number is zero only if the original number is zero, these equations yield a unique solution. This contrasts with squared equations that equal a positive number, which typically provide two distinct solutions due to the nature of squaring.

Backtracking (Reversal of Operations)

In algebra, backtracking refers to the process of reversing the operations applied to a variable in order to isolate and solve for it. This involves undoing the sequence of operations—such as multiplication, addition, exponentiation—by applying the corresponding inverse operations step by step in reverse order until the original variable is isolated.

Transcript

00:01 For this question, it's required to solve the following equation.

00:04 However, it's clear that this equation is not a quadratic equation in order to be able to solve it.

00:09 So what we can do here is to rewrite it to be in a quadratic equation form.

00:18 How, you can see that the leading term 8 .2 over 2 over 3 is actually the square of the middle term 8 .1 over 3.

00:28 So, a to 4 2 over 3 equals to a to 1 over 3 to bar 1 of 2.

00:45 So we can use this fact in order to make a proper substitution.

00:50 This substitution will be y equals to the middle term, 1 over 3.

00:55 And hence, then this will be substituted by y.

01:00 And 8th part of 2 over 3 will be equals to y with part of 2.

01:11 So what's needed now is to substitute the y squared and y inside the equation.

01:17 And notice that 3 is in the right -hand side, so i want it to be in the left -hand side in order to get the general form of the quadratic equation and to be able to solve it using the factor way.

01:29 So 8 of 2 over 3, now it becomes y -square.

01:32 2 minus 2 y and 3 will become in the other side and by doing this will be negative so it's minus 3 equals 0 so this is the quadratic equation now i want to solve it using the factor in factoring so you i want two numbers that if i multiply them together i'll get minus 3 and adding them together i'll get minus 2 so the factor would be in this form.

02:09 These are the two factors.

02:12 I have y here and y here.

02:16 You see, the sign is minus.

02:20 It means that the two numbers in these factor will have different signs.

02:30 So one is positive and the other one is negative.

02:34 And looking here, the sign is, negative so it means that the larger number between these two will have the negative sign so let's look at what possible factors of three the only two numbers that can be multiplied by each other and we get this three is three and one so based on the distribution of signs i said that the higher the greater number will get the negative sign so it's it's 3 and the lower number will get 1 and will get the positive.

03:17 So it's 1.

03:19 1 multiplied by minus 3 is minus 3 and 1 minus 3 will be minus 2...

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