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Quadratic Functions and Equations

Solving Quadratic Equations


Quadratic Equation

In standard form, quadratic equations take the form and can be solved in various ways. In general, they are solved for the value(s) of that make the equation equal to Thus, becomes Graphically, all points with a -coordinate of are the -intercepts of the function — or the zeros of the parabola. That means, solving a quadratic equation leads to finding the zeros of the parabola. Since a parabola can have or zeros, a quadratic equation can have or solutions.


Solving a Quadratic Equation Graphically

A quadratic equation can be solved graphically by expressing the equation as a function and identifying the solutions in the function's graph. Consider the following equation as an example. This and other quadratic equations can be solved using this method.


If needed, rearrange the equation and simplify

If neither side of the equation equals rearrange it so that all terms are on the same side. If there is more than one term of the same degree, simplify the equation by combining like terms.


Plot the function

The side with the variables can now be seen as a function, Plot that function in a coordinate plane. For the equation the function is


Identify any points with -coordinate

Now, find all the points on the graph that have the -coordinate


Identify the -coordinates

The -coordinates of any identified points solve the equation,

In the example, the -coordinates are and Note that solving an equation graphically does not necessarily lead to an exact answer. To verify a solution, substitute it into and evaluate the expression. Verifying the solution is done by evaluating

We find that the function value is Therefore, is an exact solution. The solution can now be verified the same way.
Since then is also a solution. Thus, the equation has two solutions and they are


Simple Quadratic Equation

Simple quadratic equations take the form and are solved using inverse operations. Once remains on the left-hand side, the equation can be written as where . The value of gives the number of solutions the equation has.


Solving Quadratic Equations with Square Roots

Quadratic equations of the form can be solved using inverse operations. To undo the exponent of on the term, square roots must be used. Consider


To begin, isolate using inverse operations to move and then to the opposite side of the equation. Here, that means adding and then dividing by


Square-root both sides of the equation
Now that has been isolated, it is necessary to undo the exponent. Exponents and radicals of the same index undo each other. Thus, square roots undo powers of and cube roots undo powers of etc. To finish isolating square-root both sides of the equation.
Using a calculator to determine gives one value — This is because calculators give the principal root of a square root. It is necessary to remember that there are two values that, when raised to the power of equal Thus, and


Factoring a Quadratic Trinomial

When factoring an expression in the form it can be difficult to see its factors. This expression can be factored by finding a pair of integers whose product is which here is and whose sum is which in the example is


List all pairs of integers whose product is equal to

The first step is to find all possible pairs of integers that multiply to In this case, and Thus, their product is To find all factor pairs, start with the pair where one factor is The other factor must then be Then continue with the pair where one of the factors is and so forth. In this case, there are three pairs.


Find the pair whose sum is

If the given expression is factorable, one of the factor pairs will add to equal In this case, Here, and is the only factor pair that adds to


Write as a sum

Now, use the factor pair to rewrite the -term of the original expression as a sum. Since the factor pair is and the middle term can be written as This gives the following equivalent expression. Notice that the expression hasn't been changed. Rather, it has been rewritten.


Factor out the GCF in the first two terms

Now, the expression has four terms, which can be grouped into the first two terms and the last two terms. Then, the GCF of each group can be factored out. Here, begin with the first group of terms, and Notice how each can be written as a product of its factors. The GCF is Therefore, it's possible to factor out


Factor out the GCF in the last two terms

Next, repeat the same process with the last two terms. In this case, and do not have any common factors, but it's always possible to write expressions as a product of and itself.


Factor out the common factor

If all previous steps have been performed correctly, there should now be two terms with a common factor, which can be seen as a repeated parentheses. Factoring out of both terms gives This means that can be written in factored form as


Solve the following equation by factoring.

Show Solution
We'll focus on the left-hand side of the equation before solving for Notice that and are the factors that multiply to equal and add to
Notice that, in factored form, the numbers in the parentheses with are the numbers from the chosen factor pair. This is because the coefficient of is We could have written the expression in factored form immediately after finding the factor pair. Lastly, we can use the zero product property to solve the equation. Thus, the solutions to the equation are and
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