Quadratic Equations and Inequalities - Algebra 2

Add 7 to both sides:

Divide both sides by the coefficient on x^2:

Add to both sides:

Form the perfect square on the left side:

Simplify the right side:

Take the square root of both sides:

Solve for x:

To complete the square, we must remember that our goal is to make a perfect square trinomial out of the terms we have.

We are given a function that we must set equal to zero if we want to find its roots:

Now, subratct 1 to the other side so we have only x-terms on one side:

Now, on the left side of the equation, in order to make a perfect square trinomial, we must take the coefficient of x - in this case, -6, and divide it by two, and then square that number:

This term becomes our "c" for the trinomial . However, because we introduced this new term on the left side of the equation, we must add it to the right hand side as well, so that we aren't "changing" the original equation:

Next, we can convert the perfect square trinomial into the square of a binomial:

This comes from the definition of the binomial, squared. When we FOIL (or use the memory tool "square the first term, square the last term, multiply the two terms and double") we get our original trinomial.

Now, to solve for x, take the square root of both sides, and add three to the other side:

.

To complete the square for any expression in the form , you must add .

In this case,

To complete the square for any expression in the form , you must add .

In this case,

In general, the discriminant is .

In this particual case .

Plug in these three values and simplify:

Given the quadratic equation of

The formula for the discriminant is (remember this as a part of the quadratic formula?)

Plugging in values to the discriminant equation:

So the discriminant is 57. What does that mean for our solutions? Since it is a positive number, we know that we will have 2 real solutions. So the answer is:

57, 2 real solutions

Remember the quadratic formula:

.

The discriminant in the quadratic formula is the term that appears under the square root symbol. It tells us about the nature of the roots.

So, to find the discriminant, all we need to do is compute for our equation, where .

We get .

The "discriminant" is the name given to the expression that appears under the square root (radical) sign in the quadratic formula, where , , and are the numbers in the general form of a quadratic trinomial: . If the discriminant is positive, the equation has real roots, and if it is negative, we have imaginary roots. In this case, , , and , so the discriminant is , and because it is negative, this equation's roots are not real.

The discriminant is found using the equation . So for the function , ,, and . Therefore the equation becomes .

The number of roots can be found by looking at the discriminant. The discriminant is determined by . For this function, ,, and . Therfore, . When the discriminant is positive, there are two real solutions to the function.

Given that quadratics can be written as . The discriminant can be found by looking at or the value under the radical of the quadratic formula. Using substiution and order of operations we can find this value of the discriminant of this quadratic equation.

The discrimiant will determine how many solutions a quadratic has. If the discriminant is positive, then there are two real solutions. If it is negative then there are two immaginary solutions. If it is equal to zero then there is one repeated solution.

Given that quadratics can be written as . The discriminant can be found by looking at or the value under the radical of the quadratic formula. Using substiution and order of operations we can find this value of the discriminant of this quadratic equation.

The discriminant is positive; therefore, there are two real solutions to this quadratic.

This is using the discriminant to find roots. The discriminant as you recall is

If you get a negative number you have no real roots, if you get zero you have one, and if you get a positive number you have two real roots.

So plug in your numbers:

Because you get a negative number you have zero real roots.

The discriminant is part of the quadratic formula. In the quadratic formula,

The discriminant is the term:

If the discriminant is 0, there is only one real solution. This would be:

, since the our discriminant is gone.

If the discriminant is a positive number, then we have two real roots, the usual form of the quadratic equation:

Finally, if the discriminant is negative, we would be taking the square root of a negative number. This will give us no real zeros.

Plugging the numbers into the discriminant gives us:

The discriminant is zero, so there is only one root,

The discriminant is found using the following formula:

For the particular function in question the variable are as follows.

Therefore:

Using the discriminant, which for a polynomial

is equal to

,

we can determine the number of roots a polynomial has. If the discriminant is positive, then the polynomial has two real roots. If it is equal to zero, the polynomial has one real root. If it is negative, then the polynomial has two roots which are complex conjugates of one another.

For our function, we have

,

so when we plug these into the discriminant formula, we get

So, our polynomial has two real roots.

To determine the amount of roots a given quadratic function has, we must find the discriminant, which for

is equal to

If d is negative, then we have two roots that are complex conjugates of one another. If d is positive, than we have two real roots, and if d is equal to zero, then we have only one real root.

Using our function and the formula above, we get

Thus, the function has only one real root.

Write the formula for the discriminant. This is the term inside the square root of the quadratic formula.

The given equation is already in the form of .

Substitute the terms into the formula.

The answer is:

The polynomial is written in the form , where

Write the formula for the discriminant. This is the term inside the square root value of the quadratic equation.

Substitute all the knowns into this equation.

The answer is: