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## Digital SAT Math

### Unit 12: Lesson 7

Solving quadratic equations: advanced# Solving quadratic equations | Lesson

A guide to solving quadratic equations on the digital SAT

## What are quadratic equations?

A

**quadratic equation**is an equation with a as its highest power term. For example, in the quadratic equation 3, x, squared, minus, 5, x, minus, 2, equals, 0:- x is the
**variable**, which represents a number whose value we don't know yet. - The squared is the
**power**or**exponent**. An exponent of 2 means the variable is . - 3 and minus, 5 are the
**coefficients**, or constant multiples of x, squared and x. 3, x, squared is a single , as is minus, 5, x. - minus, 2 is a
**constant**term.

In this lesson, we'll learn to:

- Solve quadratic equations in several different ways
- Determine the number of solutions to a quadratic equation without solving

**You can learn anything. Let's do this!**

## How do I solve quadratic equations using square roots?

### Solving quadratics by taking square roots

### When can I solve by taking square roots?

Quadratic equations without x-terms such as 2, x, squared, equals, 32 can be solved

*without*setting a quadratic expression equal to 0. Instead, we can isolate x, squared and use the square root operation to solve for x.When solving quadratic equations by taking square roots,

**both the positive and negative square roots are solutions to the equation**. This is because when we*square*a solution, the result is*always positive*.For example, for the equation x, squared, equals, 4, both 2 and minus, 2 are solutions:

- 2, squared, equals, start superscript, \checkmark, end superscript, 4
- left parenthesis, minus, 2, right parenthesis, squared, equals, start superscript, \checkmark, end superscript, 4

When solving quadratic equations without x-terms:

- Isolate x, squared.
- Take the square root of both sides of the equation. Both the positive and negative square roots are solutions.

**Example:**What values of x satisfy the equation 2, x, squared, equals, 18 ?

### Try it!

## What is the zero product property, and how do I use it to solve quadratic equations?

### Zero product property

### Zero product property and factored quadratic equations

The

**zero product property**states that if a, b, equals, 0, then either a or b is equal to 0.The zero product property lets us solve factored quadratic equations by solving two linear equations. For a quadratic equation such as left parenthesis, x, minus, 5, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, equals, 0, we know that either x, minus, 5, equals, 0 or x, plus, 2, equals, 0. Solving these two linear equations gives us the two solutions to the quadratic equation.

To solve a factored quadratic equation using the zero product property:

- Set each factor equal to 0.
- Solve the equations from Step 1. The solutions to the linear equations are also solutions to the quadratic equation.

**Example:**What are the solutions to the equation left parenthesis, x, minus, 4, right parenthesis, left parenthesis, 3, x, plus, 1, right parenthesis, equals, 0 ?

### Try it!

## How do I solve quadratic equations by factoring?

### Solving quadratics by factoring

### Solving factorable quadratic equations

If we can write a quadratic expression as the product of two linear expressions (factors), then we can use those linear expressions to calculate the solutions to the quadratic equation.

In this lesson, we'll focus on factorable quadratic equations with 1 as the coefficient of the x, squared term, such as x, squared, minus, 2, x, minus, 3, equals, 0. For more advanced factoring techniques, including special factoring and factoring quadratic expressions with x, squared coefficients other than 1, check out the

**Factoring quadratic and polynomial expressions**lesson.Recognizing factors of quadratic expressions takes practice. The factors will be in the form left parenthesis, x, plus, a, right parenthesis, left parenthesis, x, plus, b, right parenthesis, where a and b fulfill the following criteria:

- The
*sum*of a and b is equal to the coefficient of the x-term in the unfactored quadratic expression. - The
*product*of a and b is equal to the constant term of the unfactored quadratic expression.

For example, we can solve the equation x, squared, minus, 2, x, minus, 3, equals, 0 by factoring x, squared, minus, 2, x, minus, 3 into left parenthesis, x, plus, a, right parenthesis, left parenthesis, x, plus, b, right parenthesis, where:

- a, plus, b is equal to the coefficient of the x-term, minus, 2.
- a, b is equal to the constant term, minus, 3.

minus, 3 and 1 would work:

- minus, 3, plus, 1, equals, minus, 2
- left parenthesis, minus, 3, right parenthesis, left parenthesis, 1, right parenthesis, equals, minus, 3

This means we can rewrite x, squared, minus, 2, x, minus, 3, equals, 0 as left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, equals, 0 and solve the quadratic equation using the zero product property. Keep mind that a and b are

*not*themselves solutions to the quadratic equation!When solving factorable quadratic equations in the form x, squared, plus, b, x, plus, c, equals, 0:

- Rewrite the quadratic expression as the product of two factors. The two factors are linear expressions with an x-term and a constant term. The sum of the constant terms is equal to b, and the product of the constant terms is equal to c.
- Set each factor equal to 0.
- Solve the equations from Step 2. The solutions to the linear equations are also solutions to the quadratic equation.

**Example:**What are the solutions to the equation x, squared, plus, 4, x, minus, 5, equals, 0 ?

### Try it!

## How do I use the quadratic formula?

### The quadratic formula

### Using the quadratic formula to solve equations and determine the number of solutions

Not all quadratic expressions are factorable, and not all factorable quadratic expressions are easy to factor.

**The quadratic formula**gives us a way to solve any quadratic equation as long as we can plug the correct values into the formula and evaluate.For start color #7854ab, a, end color #7854ab, x, squared, plus, start color #ca337c, b, end color #ca337c, x, plus, start color #208170, c, end color #208170, equals, 0:

**Note:**the quadratic formula is

*not*provided in the reference section of the SAT! You'll have to memorize the formula to use it.

### What are the steps?

To solve a quadratic equation using the quadratic formula:

- Rewrite the equation in the form a, x, squared, plus, b, x, plus, c, equals, 0.
- Substitute the values of a, b, and c into the quadratic formula, shown below.

- Evaluate x.

**Example:**What are the solutions to the equation x, squared, minus, 6, x, equals, 9 ?

The b, squared, minus, 4, a, c portion of the quadratic formula is called the

**discriminant**. The value of b, minus, 4, a, c tells us the**number of unique real solutions**the equation has:- If b, squared, minus, 4, a, c, is greater than, 0, then square root of, b, squared, minus, 4, a, c, end square root is a
**real number**, and the quadratic equation has**two real solutions**, start fraction, minus, b, minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction and start fraction, minus, b, plus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction. - If b, squared, minus, 4, a, c, equals, 0, then square root of, b, squared, minus, 4, a, c, end square root is also 0, and the quadratic formula simplifies to start fraction, minus, b, divided by, 2, a, end fraction, which means the quadratic equation has
**one real solution**. - If b, squared, minus, 4, a, c, is less than, 0, then square root of, b, squared, minus, 4, a, c, end square root is an
**imaginary number**, which means the quadratic equation has**no real solutions**.

### Try it!

## Your turn!

## Things to remember

For a, x, squared, plus, b, x, plus, c, equals, 0:

- If b, squared, minus, 4, a, c, is greater than, 0, then the equation has 2 unique real solutions.
- If b, squared, minus, 4, a, c, equals, 0, then the equation has 1 unique real solution.
- If b, squared, minus, 4, a, c, is less than, 0, then the equation has no real solution.

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