Step 4 Check the solution in the original equation. This applies the above theorem, which says that at least one of the factors must have a value of zero. Since we have (x - 6)(x + 1) = 0, we know that x - 6 = 0 or x + 1 = 0, in which case x = 6 or x = - 1. Step 3 Set each factor equal to zero and solve for x. Solution Step 1 Put the equation in standard form. Of course, both of the numbers can be zero since (0)(0) = 0. We can never multiply two numbers and obtain an answer of zero unless at least one of the numbers is zero. We will not attempt to prove this theorem but note carefully what it states. In other words, if the product of two factors is zero, then at least one of the factors is zero. The method of solving by factoring is based on a simple theorem. This method cannot always be used, because not all polynomials are factorable, but it is used whenever factoring is possible. The simplest method of solving quadratics is by factoring.
It is possible that the two solutions are equal.Ī quadratic equation will have two solutions because it is of degree two. This theorem is proved in most college algebra books.Īn important theorem, which cannot be proved at the level of this text, states "Every polynomial equation of degree n has exactly n roots." Using this fact tells us that quadratic equations will always have two solutions. The solution to an equation is sometimes referred to as the root of the equation. In other words, the standard form represents all quadratic equations. The standard form of a quadratic equation is ax 2 + bx + c = 0 when a ≠ 0 and a, b, and c are real numbers.Īll quadratic equations can be put in standard form, and any equation that can be put in standard form is a quadratic equation.
We especially designed this trinomial to be a perfect square so that this step would work: Now rewrite the perfect square trinomial as the square of the two binomial factors That is 5/2 which is 25/4 when it is squared Now we complete the square by dividing the x-term by 2 and adding the square of that to both sides of the equation. X² + 5x = 3/4 → I prefer this way of doing it Or, you can divide EVERY term by 4 to get ĭivide through the x² term and x term by 4 to factor it out So, we have to divide the x² AND the x terms by 4 to bring the coefficient of x² down to 1. In the example following rule 2 that we were supposed to try, the coefficient of x² is 4.
As shown in rule 2, you have to divide by the value of a (which is 4 in your case). You are correct that you cannot get rid of it by adding or subtracting it out. This would be the same as rule 2 (and everything after that) in the article above.