Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Without expanding any brackets, show how to work out the exact solutions of [tex]\( 4(x+7)^2 = 25 \)[/tex].

Give the solutions.


Sagot :

Sure! Let's solve the equation step-by-step:

The given equation is:
[tex]\[ 4(x + 7)^2 = 25 \][/tex]

1. Divide both sides by 4 to isolate the squared expression:
[tex]\[ (x + 7)^2 = \frac{25}{4} \][/tex]
[tex]\[ (x + 7)^2 = 6.25 \][/tex]

2. Take the square root of both sides to remove the square:
[tex]\[ x + 7 = \pm \sqrt{6.25} \][/tex]

Since [tex]\(\sqrt{6.25} = 2.5\)[/tex], we have:
[tex]\[ x + 7 = \pm 2.5 \][/tex]

3. Solve for [tex]\(x\)[/tex] by considering the two possible cases:

- For the positive root:
[tex]\[ x + 7 = 2.5 \][/tex]
[tex]\[ x = 2.5 - 7 \][/tex]
[tex]\[ x = -4.5 \][/tex]

- For the negative root:
[tex]\[ x + 7 = -2.5 \][/tex]
[tex]\[ x = -2.5 - 7 \][/tex]
[tex]\[ x = -9.5 \][/tex]

So, the solutions to the equation [tex]\( 4(x + 7)^2 = 25 \)[/tex] are:
[tex]\[ x = -4.5 \][/tex]
[tex]\[ x = -9.5 \][/tex]