Solving Quadratic Equations - Math

Use the quadratic formula to solve:

To find the vertex of a parabola, we must put the equation into the vertex form:

The vertex can then be found with the coordinates (h, k).

To put the parabola's equation into vertex form, you have to complete the square. Completing the square just means adding the same number to both sides of the equation -- which, remember, doesn't change the value of the equation -- in order to create a perfect square.

Start with the original equation:

Put all of the terms on one side:

Now we know that we have to add something to both sides in order to create a perfect square:

In this case, we need to add 4 on both sides so that the right-hand side of the equation factors neatly.

Now we factor:

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Once we isolate , we have the equation in vertex form:

Thus, the parabola's vertex can be found at .

Begin by dividing the equation by and subtracting from each side:

Square the value in front of the and add to each side:

Factor the left side of the equation:

Take the square root of both sides and simplify:

Factor and solve:

Factor like terms:

Combine like terms:

Begin by dividing the equation by and adding to each side:

Square the value in front of the and add to each side:

Factor the left side of the equation:

Take the square root of both sides and simplify:

Begin by dividing the equation by and subtracting from each side:

Square the value in front of the and add to each side:

Factor the left side of the equation:

Take the square root of both sides and simplify:

Factor the equation to . Set both equal to zero and you get and . Remember, the zeros of an equation are wherever the function crosses the -axis.

Factor out an from the equation so that you have . Set and equal to . Your roots are and .

Set equal to zero and you get . Set equal to zero as well and you get and because when you take a square root, your answer will be positive and negative.

Factor out a from the entire equation. After that, you get . Factor the expression to . Set both of those equal to zero and your answers are and .

This expression is the difference of perfect squares. Therefore, it factors to. Set both of those equal to zero and your answers are and .

Factor the equation to . Set both equal to and you get and .

Factor a out of the quation to get

which can be further factored to

.

Set the last two expressions equal to zero and you get and .

Set each expression equal to zero and you get 0 and 6.

Set both expressions equal to . The first factor yields . The second factor gives you .

Set both expressions to and you get and .

We can factor by determining the terms that will multiply to –8 and add to +7.

Our factors are +8 and –1.

Now we can set each factor equal to zero and solve for the root.

We know that one term has a coefficient of 2 and that our factors must multiply to –10.

Our factors are +2 and –5.

Now we can set each factor equal to zero and solve for the root.

First, we can factor an term out of all of the values.

We can factor remaining polynomial by determining the terms that will multiply to +4 and add to +4.

Our factors are +2 and +2.

Now we can set each factor equal to zero and solve for the root.

Multiply both sides of the equation by , to get

This can be factored into the form

So we must solve

and

to get the solutions.

The solutions are:

and their sum is .