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Chapter 10: Problem 7

Solve the following quadratic equations. \(4 m^{2}=36\)

Short Answer

Expert verified
The solutions are \( m = 3 \) and \( m = -3 \).

Step by step solution

01

Rewrite the equation in standard form

Start by rewriting the given quadratic equation in the form of \( ax^2 + bx + c = 0 \). The given equation is \( 4 m^2 = 36\).
02

Move all terms to one side

Subtract 36 from both sides to get: \[ 4 m^2 - 36 = 0 \]
03

Factor the left side of the equation

Recognize that the left-hand side is a difference of squares: \[ (2m)^2 - 6^2 = 0 \]. Factor it as: \[ (2m - 6)(2m + 6) = 0 \]
04

Set each factor equal to zero

Set each factor equal to zero and solve for \( m \): \( 2m - 6 = 0 \) and \( 2m + 6 = 0 \)
05

Solve for \( m \)

Solve the equations: \( 2m - 6 = 0 \) gives \( m = 3 \), and \( 2m + 6 = 0 \) gives \( m = -3 \)

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Key Concepts

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

quadratic equations
Quadratic equations are polynomial equations of the second degree. This means they have the form y = ax^2 + bx + c, where a, b, and c are constants and x represents an unknown variable. These equations often have two solutions. For example, in our exercise, 4m^2 = 36, we need to rearrange it into the standard quadratic form. Once rearranged, we can solve for the variable using different methods. Quadratic equations frequently appear in physics, engineering, and various branches of mathematics, representing parabolic shapes on a graph.
factoring
Factoring is a method used to solve quadratic equations by expressing the equation as a product of its factors. Factors are simpler expressions that, when multiplied together, give the original equation. In our given equation 4m^2 - 36 = 0, the left side can be factored into (2m - 6)(2m + 6) = 0. This tells us that finding the values of m that make each factor equal to zero will solve the equation.
difference of squares
The difference of squares is a specific pattern of factoring used for terms that are squares of variables or constants subtracted from each other. It follows the formula, a^2 - b^2 = (a - b)(a + b). In our equation 4m^2 - 36 = 0, we recognize 4m^2 and 36 as perfect squares. We factor this as (2m - 6)(2m + 6) = 0. Recognizing and applying the difference of squares helps simplify and solve quadratic equations quickly.

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Most popular questions from this chapter

Solve by using the Quadratic Formula. \(16 y^{2}+8 y+1=0\)

Determine the number of solutions to each quadratic equation. (a) \(9 v^{2}-15 v+25=0\) \(100 w^{2}+60 w+9=0\) \(5 c^{2}+7 c-10=0\) \(15 d^{2}-4 d+8=0\)

A city planner wants to build a bridge across a lake in a park. To find the length of the bridge, he makes a right triangle with one leg and the hypotenuse on land and the bridge as the other leg. The length of the hypotenuse is 340 feet and the leg is 160 feet. Find the length of the bridge.

A bullet is fired straight up from a BB gun with initial velocity 1120 feet per second at an initial height of 8 feet. Use the formula \(h=-16 t^{2}+v_{0} t+8\) to determine how many seconds it will take for the bullet to hit the ground. (That is, when will \(h=0\) ?)

An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation \(\frac{1}{2} h^{2}+3 h=140\) for \(h,\) the height of the window.
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