# Precalculus : Solving Quadratic Equations

## Example Questions

### Example Question #4 : Solve A Quadratic Equation

Solve the quadratic equation for .

Explanation:

There are two solutions; .

We proceed as follows.

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.

### Example Question #5 : Solve A Quadratic Equation

Solve the quadratic equation for .

Explanation:

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

Plugging in our given values  we have:

### Example Question #6 : Solve A Quadratic Equation

Find the roots of the equation.

Explanation:

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.

let

### Example Question #7 : Solve A Quadratic Equation

Solve .

Explanation:

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: .

### Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

True or false: for a quadratic function of form ax+ bx + c = 0, if the discriminant b- 4ac = 0, there is exactly one real root.

False

True

True

Explanation:

This is true. The discriminant b- 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 b- 4ac and evaluate, three cases can happen:

b- 4ac > 0

b- 4ac = 0

b- 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 b- 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.

### Example Question #2 : Find Roots Of Quadratic Equation Using Discriminant

True or false: for a quadratic function of form ax+ bx + c = 0, if the discriminant b- 4ac > 0, there are exactly 2 distinct real roots of the equation.

True

False

True

Explanation:

This is true. The discriminant b- 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 b- 4ac and evaluate, three cases can happen:

b- 4ac > 0

b- 4ac = 0

b- 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 b- 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.

### Example Question #3 : Find Roots Of Quadratic Equation Using Discriminant

True or false: for a quadratic function of form ax+ bx + c = 0, if the discriminant b- 4ac < 0, there are exactly two distinct real roots.

False

True

False

Explanation:

This is false. The discriminant b- 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 b- 4ac and evaluate, three cases can happen:

b- 4ac > 0

b- 4ac = 0

b- 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 b- 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.

### Example Question #4 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b2 - 4ac to find the discriminant of the following equation: 4x2 + 19x - 5 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Discriminant: 281

Two imaginary roots:

Discriminant: 441

Two real roots:  or

Discriminant: 441

Two real roots:  or

Discriminant: 0

One real root:

Discriminant: 281

Two imaginary roots:

Discriminant: 441

Two real roots:  or

Explanation:

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

or

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

### Example Question #5 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b2 - 4ac to find the discriminant of the following equation: 4x2 + 12x + 10 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Discriminant: 16

Types of Roots: Two distinct real roots

Exact Roots: -1, -2

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots:

Discriminant: 304

Types of Roots: Two distinct real roots

Exact Roots:

Discriminant: 304

Types of Roots: Two distinct real roots

Exact Roots:

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots:

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots:

Explanation:

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

### Example Question #6 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b2 - 4ac to find the discriminant of the following equation: -3x2 + 6x - 3 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Discriminant: 0

One real root: x = -1

Discriminant: 72

Two distinct real roots:

Discriminant: 72

Two distinct real roots:

Discriminant: 0

One real root: x = 1

Discriminant: -72

Two distinct imaginary roots:

Discriminant: 0

One real root: x = 1

Explanation:

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.