Save my name, email, and website in this browser for the next time I comment. Toggle navigation. Search Log In. To do 3 min read. Factor out the greatest common factor. Divide both sides by 3 to isolate the trinomial. Subtract 5 to get the constant terms on the right.
Take half of 4 and square it. Factor the perfect square trinomial as a binomial square. Use the Square Root Property. Solve for x. Rewrite to show 2 solutions. Add the fractions on the right side. Let's complete the square on the general equation and see exactly how that produces the Quadratic Formula.
Recall the process of completing the square. Try it yourself before you continue to the example below. Hint: Notice that in the general equation, the coefficient of x 2 is not equal to 1. Divide both sides of the equation by a , so that the coefficient of x 2 is 1. Since the coefficient on x is , the value to add to both sides is. Write the left side as a binomial squared. Evaluate as. Write the fractions on the right side using a common denominator.
Add the fractions on the right. Use the Square Root Property. Remember that you want both the positive and negative square roots! Subtract from both sides to isolate x. The denominator under the radical is a perfect square, so:. Add the fractions since they have a common denominator. There you have it, the Quadratic Formula. Solving a Quadratic Equation using the Quadratic Formula. The Quadratic Formula will work with any quadratic equation, but only if the equation is in standard form,.
To use it, follow these steps. Be careful to include negative signs if the bx or c terms are subtracted. That's a lot of steps. Use the Quadratic Formula to solve the equation.
First write the equation in standard form. Note that the subtraction sign means the constant c is negative. Substitute the values into the Quadratic Formula. Simplify, being careful to get the signs correct. Simplify the radical:. Separate and simplify to find the solutions to the quadratic equation. Note that in one, 6 is added and in the other, 6 is subtracted. The power of the Quadratic Formula is that it can be used to solve any quadratic equation, even those where finding number combinations will not work.
Most of the quadratic equations you've looked at have two solutions, like the one above. The following example is a little different. Subtract 6 x from each side and add 16 to both sides to put the equation in standard form. Identify the coefficients a , b , and c. Since 8 x is subtracted, b is negative. Since the square root of 0 is 0, and both adding and subtracting 0 give the same result, there is only one possible value.
Again, check using the original equation. Let's try one final example. This one also has a difference in the solution. Simplify the radical, but notice that the number under the radical symbol is negative! Check these solutions in the original equation. Be careful when expanding the squares and replacing i 2 with You may have incorrectly factored the left side as x — 2 2. The correct answer is or. Using the formula,. If you forget that the denominator is under both terms in the numerator, you might get or.
However, the correct simplification is , so the answer is or. The Discriminant. These examples have shown that a quadratic equation may have two real solutions, one real solution, or two complex solutions.
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