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How do you calculate an equation?
Solving equations involves finding the unknowns in the equation. Let’s start with simple equations. You have an equation with one unknown – call it x. The trick here to solving the equation is to end up with x on one side of the equation and a number on the other. You do this by adding, subtracting, multiplying or dividing both sides of the equation. Remember, whatever you do to one side of the equation, you must do the same to the other side.
A quadratic equation is a second-degree polynomial having the general form ax²+bx+c=0, where a, b, and c are constants. There are multiple methods to solve quadratics: factorization, completing the square, and the quadratic formula.
First up is factorization. How do you factorize a quadratic? The trick is to get the equation to the form (x-u)(x-v)=0, now we have to solve much simpler equations.
Solving quadratics by factorizing usually works just fine. But what if the quadratic equation can’t be factored, you’re going to need a different method to help you solve it, completing the square.
An equation in which one side is a perfect square trinomial can be easily solved by taking the square root of each side. Easy is good, so we basically want to force the quadratic equation into the form (x+a)²=x²+2ax+a².
All it takes is making sure that the coefficient of the highest power (x²) is one. Move the constant term to the right hand side. Take half of the coefficient of the middle term(x), square it, and add that value to both sides of the equation. Factor the perfect square trinomial. Take the square root of each side and solve.
To make things simple, a general formula can be derived such that for a quadratic equation of the form ax²+bx+c=0 the solutions are x=(-b ± sqrt(b^2-4ac))/2a. The quadratic formula comes in handy, all you need to do is to plug in the coefficients and the constants (a,b and c).
Solving exponential equations is straightforward; there are basically two techniques:
- If the exponents on both sides of the equation have the same base, you can use the fact that: If a^x=a^y then x=y. Now simply solve the equation.
- In all other cases, take the log of both sides (this might require some manipulation) and solve for the variable. The logarithm property ln(a^x)=xln(a) makes this a fairly simple task.
Radical equations are equations involving radicals of any order. To solve radical equations, you first have to get rid of the radicals, in the case of square roots square both sides of the equation (in some cases this should be done multiple times), then simplify the new equation (either linear or quadratic) and solve. One more thing to note, by squaring the equation we changed the original equation, so it is very important to check the solutions at the end.
The key to solving equations is to identify the equation type. Other equation types to know are the biquadratic, rational, logarithmic, and absolute.
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