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Chapter 9: Problem 61

Solve using the Square Root Property. $$(m-4)^{2}+3=15$$

Short Answer

Expert verified
m = 4 \pm 2\sqrt{3}

Step by step solution

01

Isolate the squared term

First, subtract 3 from both sides of the equation to isolate the squared term \( (m-4)^2 + 3 = 15 \, (m-4)^2 = 12 \).
02

Apply the Square Root Property

Next, take the square root of both sides. Remember to consider both the positive and negative square roots: \( \sqrt{(m-4)^2} = \pm\sqrt{12} \, m-4 = \pm 2\sqrt{3} \).
03

Solve for the variable

Finally, solve for m by adding 4 to both sides: \( m - 4 = 2\sqrt{3} \, m = 4 + 2\sqrt{3} \) and \( m - 4 = -2\sqrt{3} \, m = 4 - 2\sqrt{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Property
The Square Root Property is a useful tool for solving quadratic equations, especially when the equation is in the form (something)虏 = (something else). This property tells us that if we have an equation like (x - a)^2 = b we can solve it by taking the square root of both sides. When you take the square root of both sides, we need to remember two important things:
  • Consider both the positive and negative roots.
  • This means your equation becomes x - a = 卤鈭歜.
In our example, we had (m-4)虏 + 3 = 15. First, we isolated the squared term (m-4)虏 by subtracting 3 from both sides:
  • (m-4)虏 = 12
Next, we applied the Square Root Property by taking the square root of both sides:
  • 鈭�((m-4)虏) = 卤鈭�12
This leaves us with m-4 = 卤2鈭�3.
Isolating Squared Term
Isolating the squared term is an essential first step when solving quadratic equations using the Square Root Property. This means getting the term with the squared variable by itself on one side of the equation. Let's break down this process:
  • Start with your equation: (m-4)虏 + 3 = 15.
  • Subtract 3 from both sides to isolate the squared term: (m-4)虏 = 12
By isolating the squared term, you prepare the equation for the next step, which is applying the Square Root Property. Isolating the squared term helps simplify the equation, making it easier to solve for the variable.
Positive and Negative Square Roots
After isolating the squared term in a quadratic equation, the next step is to solve for the variable by applying the Square Root Property. It's important to consider both the positive and negative square roots, as a squared number has two roots: one positive and one negative. Here's how to handle positive and negative square roots in our example equation:
  • We isolated the squared term: (m-4)虏 = 12.
  • Then, we took the square root of both sides considering both positives and negatives: 鈭�((m-4)虏) = 卤鈭�12.
  • This results in two equations: m-4 = 2鈭�3 and m-4 = -2鈭�3
To solve for m, we add 4 to both sides of each equation, giving us two solutions:
  • m = 4 + 2鈭�3
  • m = 4 - 2鈭�3
By considering both positive and negative roots, we ensure that all possible solutions are accounted for. This is a crucial step in solving the quadratic equation correctly.

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Most popular questions from this chapter

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=3 x^{2}+18 x+20$$

Solve. Round answers to the nearest tenth. A stone is thrown vertically upward from a platform that is 20 feet height at a rate of \(160 \mathrm{ft} / \mathrm{sec}\) Use the quadratic function \(h(t)=-16 t^{2}+160 t+20\) to find how long it will take the stone to reach its maximum height, and then find the maximum height.

Find the maximum or minimum value of each function. $$y=4 x^{2}-49$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=x^{2}-6 x+15$$

Graph each function using transformations. $$f(x)=(x-4)^{2}-3$$
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