# Quadratic equations

Proportionally, the monitors appear very similar. Note that the two roots are irrational. Solving quadratic equations by factoring The method of solving quadratic equations by factoring rests on the simple fact, used in example 2 above, that if we obtain zero as the product of two numbers then at least one of the numbers must be zero.

We usually will go straight to the answer from the squared part.

## Solving quadratic equations

Here is the work for this equation. It is in standard form. Substitute 2 for a, -1 for b, and -1 for c in the quadratic formula and simplify. There are three basic methods of solving such quadratic equations: by factoring by completing the square by the quadratic formula Each method is important and needs to be mastered. Different situations will require different approaches, and while the last two methods always work, the method of factoring is very quick and accurate, provided the equation has rational solutions. Substituting in the quadratic formula, Since the discriminant b 2 — 4 ac is 0, the equation has one root. The two previous examples were relatively easy since in the first case it was easy to isolate the unknown while in the second, a common factor enabled the left-hand side to be easily factored. So what you will be looking for is where the graph of crosses the x-axis. Take the square root of both sides. If we were to factor the equation, we would get back the factors we multiplied.

First, simplify by putting all terms on one side and combining like terms. In this case we have a product of three terms that is zero.

It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.

Then use the quadratic formula. Show Solution The first thing to do is factor this equation as much as possible.

### Quadratic equation roots

Show Solution Take the square root of both sides, and then simplify the radical. But if you were to express the solution using imaginary numbers, the solutions would be. These two methods work just as well when the coefficient of x2 is not one. Use the numbers exactly as they are. How To: Given a quadratic equation with the leading coefficient of 1, factor it Find two numbers whose product equals c and whose sum equals b. The problem is, of course, that it is sometimes not easy to do the factoring. You can see from the graph that there are two x-intercepts, one at 1 and one at. Do you get the idea that might be bad? Use the discriminant to determine the number and type of solutions to a quadratic equation. We saw some of these back in the Solving Linear Equations section and since they can also occur with quadratic equations we should go ahead and work on to make sure that we can do them here as well. Another way of saying this is that the x-intercepts are the solutions to this equation. Substitute 2 for a, -1 for b, and -1 for c in the quadratic formula and simplify. Example 4 Solve each of the following equations.

So, technically we need to set each one equal to zero and solve. Do you get the idea that might be bad? No real root if the discriminant b 2 — 4 ac is a negative number.

### Solving quadratic equations by factoring

The discriminant tells us whether the solutions are real numbers or complex numbers as well as how many solutions of each type to expect. Pay close attention when substituting, and use parentheses when inserting a negative number. Check by inserting your answer in the original equation. This means that at least one of the following must be true. There are three basic methods of solving such quadratic equations: by factoring by completing the square by the quadratic formula Each method is important and needs to be mastered. Substituting in the quadratic formula, Since the discriminant b 2 — 4 ac is negative, this equation has no solution in the real number system. When using the quadratic formula, you should be aware of three possibilities. But if you were to express the solution using imaginary numbers, the solutions would be. So, provided we can factor a polynomial we can always use this as a solution technique. It will also arise in other sections of this chapter and even in other chapters. This symbol is shorthand that tells us that we really have two numbers here.

If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

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