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Chapter 9: Problem 7

Solve each equation. $$4 m^{2}=36$$

Short Answer

Expert verified
m = \frac{1}{3}\text{ or } \text{ } \text{ } \frac{4}{4}, \text{ } \frac{ &-3}

Step by step solution

01

Isolate the variable term

Divide both sides of the equation by 4 to isolate the term with the variable on one side. Equation: 4m^2 = 36 Divide both sides by 4: $$m^2 = \frac{36}{4} = 9$$
02

Take the square root of both sides

To solve for m, take the square root of both sides of the equation. Equation: $$m^2 = 9$$ Take the square root of both sides: $$m = \begin{pmatrix} \frac{1}{3}\text{ or } \text{ } \frac{9}{4}, \text{ } \frac{ m}{_9}$$ = -3$$

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Key Concepts

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

Solving Quadratic Equations
Quadratic equations are polynomials of degree 2, taking the form \(ax^2 + bx + c = 0\). Solving them means finding the values of the variable that make the equation true.
There are different methods for solving quadratic equations, such as:
  • Factoring
  • Using the quadratic formula\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
  • Completing the square
  • Graphing
In some cases, like in the provided exercise, taking the square root to isolate the variable term simplifies the process. Here, the quadratic equation form simplifies to \(m^2 = 9\), making it easier to solve.
Isolating Variables
Isolating variables is a crucial step in solving equations. It involves manipulating the equation to have the variable term alone on one side of the equation.
In the given example: $$4m^2 = 36$$, we divide both sides by 4 to isolate the variable term, giving: $$m^2 = 9$$.This step simplifies our equation, making subsequent steps easier.
Remember these tips:
  • Perform the same operation on both sides to keep the equation balanced.
  • Simplify each term step-by-step.
  • Combining like terms and reducing fractions helps in clear isolation of the variable.
Mastering this skill makes solving equations straightforward and efficient.
Taking Square Roots
Taking square roots is a method used to solve quadratic equations, particularly when they are in the form \(x^2 = d\).
From the isolated equation \(m^2 = 9\), taking the square root of both sides yields: \(m = \pm 3\). This shows both possible solutions since the square of both positive and negative numbers gives the same result.
Key tips:
  • Use the \(\pm\) symbol when taking square roots to consider both positive and negative roots.
  • Check both solutions in the original equation to ensure they are correct.
This step is straightforward but crucial, ensuring you find all possible solutions.

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Most popular questions from this chapter

Solve. Round answers to the nearest tenth. A ball is thrown vertically upward from the ground with an initial velocity of \(122 \mathrm{ft} / \mathrm{sec}\). Use the quadratic function \(h(t)=-16 t^{2}+122 t+0\) to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=5 x^{2}-10 x+8$$

Solve each inequality algebraically and write any solution in interval notation. $$x^{2}+x-6 \leq 0$$

(a) solve graphically and (b) write the solution in interval notation. $$x^{2}+4 x+3 \leq 0$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=2 x^{2}-4 x+1$$
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