X^y*X^z=X^yz is sometimes true. For instance, if y=3 and z=5, the answer would be X^8 because the exponents are added together. However, if y and z equal 2, added and multiplied together they equal 4. Therefore, this equation is only true if the numbers are the same when adding and multiplying them together.
Quadratic equations are equations in the format ax^2+bx+c. Ax^2 represents the quadratic term, bx represents the linear term, and c is the constant term. The graphs for these equations will always look like a U or an upside-down U.
One way to solve quadratic equations is by graphing. Taking the equation x^2+6x+8 and plugging it in give two solutions, -2 and -4, as shown by the graph below. Another way to solve quadratic equations is by factoring. Using the same equation, we must find two numbers that can be multiplied together to equal 8 and added together to equal 6. (x+2)(x+4) can be foiled out to check the work, and it equals our original equation. Because this equation is assumed to be set equal to 0, to find what x equals each parenthesis must be set equal to 0. If the first one, x+2, is equal to zero, then subtracting 4 from each side will result in one answer. When x+4=0, x=-4. The third way to solve quadratic equations is by completing the square. Using the same equation, we first subtract 8 from each side. Then, to complete the square, divide the 6 by 2. Take the quotient, square it, and add the number to each side. Now our equation(x^2+6x+9=-8+9) can be factored out. (x+3)^2=1. Square root each side to get rid of the exponent. Taking the square root of a variable makes it either plus or minus the number, so the equation is x+3=(+or-)1. Now the equation can easily be solved. x+3=1. Subtract 3 from each side and x=-2. In x+3=-1, subtract 3 from each side to get x=-4. The final method is solving using the equation x=-b(+or-) the square root of b^2-4(a)(c) all over 2a. If a=1, b=6, and c=8, then the equation is x=-6(+or-) the square root of 36-4(1)(8) all over 2(1). Simplify to get x=-6(+or-) the square root of 4 all over 2. The square root of 4 is 2. Everything in the equation is divisible by 2. x=-3(+or-)1. x=-3+1 is -2 and x=-3-1 is -4. |
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April 2015
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