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Chapter 7: Problem 79

Solve each equation. $$2 x^{2}+3=7$$

Short Answer

Expert verified
x = \pm \sqrt{2}

Step by step solution

01

Isolate the quadratic term

First, subtract 3 from both sides of the equation to isolate the term with the variable. \[2x^2 + 3 - 3 = 7 - 3\]This simplifies to:\[2x^2 = 4\]
02

Solve for the quadratic term

Divide both sides of the equation by 2 to further isolate the quadratic term.\[\frac{2x^2}{2} = \frac{4}{2}\]This simplifies to:\[x^2 = 2\]
03

Apply the square root

Take the square root of both sides of the equation to solve for x. Don't forget to include both the positive and negative roots.\[x = \pm \sqrt{2}\]

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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 a type of polynomial equation of the second degree. This means the equation will have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our case, we are working with the equation \(2x^2 + 3 = 7\). Quadratic equations can be solved in several ways, including factoring, completing the square, and using the quadratic formula. In this exercise, we use the method of isolation and square rooting to find the solution.
Isolation of Terms
The first step in solving a quadratic equation often involves isolating the variable terms. Here, we started with the equation \(2x^2 + 3 = 7\). To isolate the term containing \(x^2\), we subtracted 3 from both sides. This eliminated the constant term on the left side, simplifying it to \(2x^2 = 4\). Isolation of terms helps reduce the equation to a simpler form that makes it easier to solve for the variable.
Square Roots
Once we have isolated the quadratic term \(x^2\), we need to solve for \(x\). To do this, we took the square root of both sides of the equation \(x^2 = 2\). It's important to remember that taking the square root can result in both a positive and a negative solution. Therefore, the solutions are \(x = \pm \sqrt{2}\). Square roots are a crucial part of solving quadratic equations, especially when the equation has been reduced to the form \(x^2 = k\), where \(k\) is a constant.

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