There are two solutions; .

We proceed as follows. 

Add  to both sides.

Take the square root of both sides, remember to introduce plus/minus on the right side since you are introducing a square root into your work.

Add  to both sides.

For any quadtratic equation of the form , the quadratic formula is

Plugging in our given values  we have:

Use either the quadratic formula or factoring to solve the quadratic equation.

Using factoring, we want to find which factors of six when multiplied with the factors of two and then added together result in negative one.

Using the quadratic formula,

let 

To solve this equation, use trial and error to factor it. Since the leading coefficient is , there is only one way to get  , so that is helpful reminder. Once it's properly factored, you get: . Then, set both of those expressions equal to  to get your roots: .

This is true. The discriminant b² - 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b² - 4ac and evaluate, three cases can happen:

b² - 4ac > 0

b² - 4ac = 0

b² - 4ac < 0

In the first case, having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes , this scenario yields two real results.

In the middle case (the case of our example), . Going back to the quadratic formula  , you can see that when everything under the square root is simply 0, then you get only , which is why you have exactly one real root.

For the final case, if b² - 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

This is true. The discriminant b² - 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b² - 4ac and evaluate, three cases can happen:

b² - 4ac > 0

b² - 4ac = 0

b² - 4ac < 0

In the first case (the case of our example), having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes , this scenario yields two real results.

In the middle case, . Going back to the quadratic formula  , you can see that when everything under the square root is simply 0, then you get only , which is why you have exactly one real root.

For the final case, if b² - 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

This is false. The discriminant b² - 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b² - 4ac and evaluate, three cases can happen:

b² - 4ac > 0

b² - 4ac = 0

b² - 4ac < 0

In the first case, having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes , this scenario yields two real results.

In the middle case, . Going back to the quadratic formula  , you can see that when everything under the square root is simply 0, then you get only , which is why you have exactly one real root.

For the final case (the case of our example), if b² - 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

In the above equation, a = 4, b = 19, and c = -5. Therefore:

b² - 4ac = (19)² - 4(4)(-5) = 361 + 80 = 441. 

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two real roots. 

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

which simplifies to

which gives us two answers:

 or 

These values of x are the two distinct real roots of the given equation.

In the above equation, a = 4, b = 12, and c = 10. Therefore:

b² - 4ac = (12)² - 4(4)(10) = 144 - 160 = -16. 

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two distinct imaginary roots. 

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

In other words, our two distinct imaginary roots are  and 

In the above equation, a = -3, b = 6, and c = -3. Therefore:

b² - 4ac = (6)² - 4(-3)(-3) = 36 - 36 = 0. 

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, there is exactly one real root.

Finally, we use the quadratic function to find these exact root. The quadratic formula is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

which simplifies to

which gives us one answer: x = 1

This value of x is the one distinct real root of the given equation.