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Chapter 5: Problem 26

Solve each equation. $$ -2 x^{2}+8=0 $$

Short Answer

Expert verified
x = 2, -2

Step by step solution

01

Move constant term to the other side

Start by subtracting 8 from both sides of the equation to isolate the term with the variable. This gives: \[-2x^2 = -8\]
02

Divide by the coefficient of the squared term

Divide both sides of the equation by -2 to isolate the square term: \[x^2 = 4\]
03

Solve for x

Take the square root of both sides to solve for x. Remember to include both the positive and negative roots in your solution: \[x = \pm \sqrt{4} = \pm 2\]

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Key Concepts

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

isolating variable
When solving quadratic equations like \-2x^2 + 8 = 0, the first step is to isolate the term with the variable. This means we need to get the x^2 term by itself on one side of the equation.
To do this, we can move the constant term (which is the number without a variable) to the other side. In our example, subtracting 8 from both sides simplifies the equation to:
\[-2x^2 = -8\]
This step is crucial because it sets up the equation so that we can further isolate x.
taking square roots
Once we've isolated the square term (x^2) in the equation -2x^2 = -8, we need to simplify it further. To make things easier, divide both sides of the equation by the coefficient of x^2, which in this case is -2. Doing so gives us:
\[x^2 = 4\]
Next, we take the square root of both sides of the equation. This is a key step in solving quadratic equations because it helps us find the value of x. Taking the square root of 4 results in:
\[ \sqrt{x^2} = \sqrt{4} \rightarrow x = \pm 2\] Remember, the square root of a number has both positive and negative roots.
positive and negative roots
When we solve quadratic equations by taking square roots, it is important to remember that the solution will always have two values: one positive and one negative. This is because squaring either a positive or negative number will result in the same positive value.
In our example, after taking the square root, we have two solutions:
\[x = \sqrt{4} = 2\] and \[x = - \sqrt{4} = -2\]
Therefore, our final answer is:
\[ x = \pm 2 \rightarrow x = 2 \text{ and } x = -2 \] This duality of positive and negative roots is a fundamental concept when dealing with square roots in quadratic equations.

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