Explain in your own words how the principle of square roots is used to solve quadratic equations.

Question

 

What form must a quadratic equation be in to use the principle of square roots
for solving? Demonstrate the process using your own example or
solve the
quadratic equation x2 + 6x -16 = 0 by completing the
square. Explain how to solve a quadratic
equation using the quadratic formula. Demonstrate the process with your own
example or solve the
quadratic equation x2 + 6x -16 = 0 by using the quadratic
formula.

 

Answer 1

The principle of square roots requires that the square root of the side with the unknown (x) only includes x without any exponent (other than 1).
The simplest form that fits this criterion is:
x² = k
x = +/- sqrt(k)
but x can also be an expression. For quadratic equations, the standard form can
be used like this:

a(x-h)² + k = 0
(x-h)² = -k/a

Now you can take the square root of both sides and solve for x:

x-h = +/- sqrt(-k/a)
x = h +/- sqrt(-k/a)

Note that in order to get real numbers as answers, a and k must have opposite signs so that -k/a is positive.

An equation in the general form can be converted to the standard form:

ax² + bx + c = 0
becomes
a(x-h)² + k = 0
(x-h)² = -k/a

where:

a = a,  h = b/2a,  k = c/a - (b/2a)²

(This conversion uses "completing the square".)

It is usually easier to solve a general form equation using the more familiar factoring or the quadratic
formula.

I Hope this helps.

Gene G.

Answer 2

Harry,

The principle of square roots applies to "simple" quadratic equations, like x²-4 = 0. In this case, you can write the equation as x² = 4. By taking the square root of both sides, you get x = +/- 2.

With your example of x² + 6x - 16 = 0, add 16 to both sides. This gives you x2 + 6x = 16. You can rewrite this as x2 + 6x + __ = 16 + __ (remember that you have to add the same thing to both sides). This will allow you to factor the left side to get (x + 3)² = 16 + __ . Use the principle of square roots by taking the square root of both sides.

As for the quadratic formula, I can't put the formula in this answer, but I suggest taking a look at http://www.purplemath.com/modules/quadform.htm for the actual formula. To get you started, a = 1, b = 6, and c = -16.

Hope this helps.
Ethan