how do you solve an equation by extracting square roots? my book can't help, and the teach can't explain it so i understand it.
to take out square roots, just basically square the number. for example to take square root of four is 2 but to take out the square root is square the four and your answer remains 4
If x^2 = a, then:
x = ± sqrt[a]
If x^2 + p x + q = 0, then you want to get rid of the linear p x term. You can do that by substituting:
x = y - p/2
The linear term in y then cancels and you can solve for y by taking the square root.
Third and fourth degree equations can be solved too by extracting roots. But equations of fifth or higher degree cannot (in general) be solved by extracting roots.
Solving an equation by extracting square roots involves finding the value of the variable that satisfies the equation by isolating the variable and taking the square root of both sides. Here's a step-by-step guide to help you understand the process:
1. Ensure that the equation is in the form where the variable you want to solve is isolated on one side and the other side is a perfect square.
Example: If the equation is x² = 16, it can be solved by extracting square roots.
2. Take the square root of both sides of the equation. This means finding the number that, when squared, gives you the value on the other side of the equation.
For example, taking the square root of both sides of x² = 16 gives us √(x²) = ±√16.
3. Simplify. On the left side (√(x²)), the square and the square root will cancel out, giving you |x| (read as "absolute value of x"). On the right side (√16), the square root of 16 is 4.
4. Introduce the positive and negative possibilities by including ± on the right side of the equation, resulting in |x| = ±4.
5. Solve for x by considering both the positive and negative cases:
Case 1: |x| = 4
In this case, x can either be 4 or -4.
Case 2: |x| = -4
Since absolute values are always non-negative, there are no solutions in this case.
6. Therefore, the possible solutions to the original equation x² = 16 are x = 4 and x = -4.
Remember to check your answers by substituting them back into the original equation to ensure they satisfy the equation.