The Digital SAT Math Guide to Quadratic Equations (Part 1)

Master Quadratic Equations for the Digital SAT with Step-by-Step Explanations

Quadratic equations are one of the most important algebra topics on the Digital SAT. They appear in many forms, from straightforward equation-solving questions to graph interpretation, mathematical modeling, and real-world word problems. A strong understanding of quadratics also makes it much easier to learn functions, parabolas, coordinate geometry, and polynomial expressions.

Unlike linear equations, which produce straight lines when graphed, quadratic equations create curved graphs called parabolas. Learning how these equations behave will help you answer a wide variety of SAT Math questions quickly and accurately.

This guide is written for students who want to build a solid understanding of quadratics from the ground up. Every solution is explained one step at a time, with no skipped steps or unexplained shortcuts. By the time you finish this chapter, you'll understand what quadratic equations are, how to recognize them, and how to solve many of them by factoring.

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Learning Goals

In this chapter, you will learn how to:

• Recognize a quadratic equation.

• Understand why quadratic equations are different from linear equations.

• Identify the standard form of a quadratic equation.

• Understand quadratic expressions and quadratic functions.

• Solve simple quadratic equations by factoring.

• Apply the Zero Product Property.

• Check your answers correctly.

• Avoid common mistakes made on the Digital SAT.

These concepts form the foundation for more advanced methods such as completing the square and using the quadratic formula, which will be covered in later chapters.

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What Is a Quadratic Equation?

A quadratic equation is an equation in which the highest exponent of the variable is 2.

Examples include:

x² = 49

x² + 5x + 6 = 0

2x² − 7x + 3 = 0

4x² = 100

Notice that each equation contains x².

That squared variable is what makes the equation quadratic.

Compare these two equations.

Linear equation:

2x + 7 = 13

Highest exponent = 1

Quadratic equation:

x² + 2x − 15 = 0

Highest exponent = 2

The difference may seem small, but it changes how the equation behaves. A linear equation usually has one solution, while a quadratic equation can have two solutions, one solution, or no real solutions.

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The Standard Form of a Quadratic Equation

Most quadratic equations on the SAT are written in standard form:

ax² + bx + c = 0

Each letter has a meaning.

a is the coefficient of x².

b is the coefficient of x.

c is the constant term.

For example,

3x² + 8x − 11 = 0

Here,

a = 3

b = 8

c = −11

Learning to identify these three values is important because later methods, especially the quadratic formula, use them directly.

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Understanding the Parts of a Quadratic Equation

Consider

2x² + 9x − 18 = 0

This equation has three terms.

First term:

2x²

This is called the quadratic term because it contains x².

Second term:

9x

This is called the linear term because it contains x.

Third term:

−18

This is the constant term because it contains no variable.

Recognizing these parts helps you identify which solving method to use.

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What Does It Mean to Solve a Quadratic Equation?

Solving a quadratic equation means finding every value of the variable that makes the equation true.

For example,

x² = 25

Which numbers produce 25 when squared?

5² = 25

(−5)² = 25

Therefore,

x = 5

and

x = −5

Unlike linear equations, quadratic equations often have more than one correct answer.

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Why Are There Two Answers?

Many students are surprised to discover that one equation can have two solutions.

The reason is simple.

Squaring removes the negative sign.

Positive example:

5 × 5 = 25

Negative example:

−5 × −5 = 25

Both calculations produce the same answer.

Whenever you solve an equation involving x², always ask yourself whether both a positive and a negative solution are possible.

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Introduction to Factoring

Factoring  x² + bx + c   type  when a = 1

Factoring is one of the fastest methods for solving many quadratic equations on the Digital SAT.

Factoring means rewriting an expression as the product of two smaller expressions.

Example:

x² + 5x + 6

can be written as

(x + 2)(x + 3)

When multiplied together,

(x + 2)(x + 3)

= x² + 3x + 2x + 6

= x² + 5x + 6

The original expression and its factored form are mathematically identical.

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The Zero Product Property

Factoring works because of an important algebra rule.

If

A × B = 0

then

A = 0

or

B = 0

or both.

This rule is called the Zero Product Property.

Example:

(x + 4)(x − 7) = 0

Either

x + 4 = 0

or

x − 7 = 0

Solve each equation separately.

First equation:

x + 4 = 0

Subtract 4 from both sides.

x + 4 − 4 = 0 − 4

Simplify.

x = −4

Second equation:

x − 7 = 0

Add 7 to both sides.

x − 7 + 7 = 0 + 7

Simplify.

x = 7

Therefore,

the two solutions are

x = −4

and

x = 7

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Example 1

Solve

x² + 7x + 12 = 0

Step 1

Write the equation.

x² + 7x + 12 = 0

Step 2

Find two numbers whose product is 12 and whose sum is 7.

Possible factor pairs of 12 are

1 and 12

2 and 6

3 and 4

Only

3 and 4

add to 7.

Step 3

Write the factors.

(x + 3)(x + 4) = 0

Step 4

Apply the Zero Product Property.

Either

x + 3 = 0

or

x + 4 = 0

Step 5

Solve the first equation.

Subtract 3 from both sides.

x + 3 − 3 = 0 − 3

Simplify.

x = −3

Step 6

Solve the second equation.

Subtract 4 from both sides.

x + 4 − 4 = 0 − 4

Simplify.

x = −4

Final Answer

x = −3

x = −4

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Example 2

Solve

x² − 9x + 20 = 0

Step 1

Find two numbers whose product is 20.

1 and 20

2 and 10

4 and 5

Step 2

Which pair adds to −9?

Since the product is positive and the sum is negative,

both numbers must be negative.

−4 and −5

Step 3

Write the factors.

(x − 4)(x − 5) = 0

Step 4

Set each factor equal to zero.

x − 4 = 0

x − 5 = 0

Step 5

Solve.

Add 4 to both sides.

x = 4

Add 5 to both sides.

x = 5

Check

4² − 9(4) + 20

16 − 36 + 20

0

Correct.

Now check 5.

25 − 45 + 20

0

Correct.

Both answers satisfy the equation.

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Example 3

Solve

x² + x − 12 = 0

Step 1

Find two numbers whose product is −12.

Possible pairs include

1 and −12

2 and −6

3 and −4

Step 2

Find the pair whose sum equals 1.

4 and −3

Step 3

Write the factors.

(x + 4)(x − 3) = 0

Step 4

Set each factor equal to zero.

x + 4 = 0

x − 3 = 0

Step 5

Solve.

Subtract 4 from both sides.

x = −4

Add 3 to both sides.

x = 3

Final Answer

x = −4

x = 3

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A Quick Factoring Strategy

Whenever you see

x² + bx + c

ask yourself two questions.

Question 1

Which two numbers multiply to give c?

Question 2

Do those same numbers add to give b?

If the answer is yes,

you have found the correct factors.

With practice, this process becomes much faster.

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Forgetting to Check

Substitute every solution back into the original equation.

If the equation balances,

your solution is correct.

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Practice Questions

Solve by factoring.

1. x² + 5x + 6 = 0

2. x² − 8x + 15 = 0

3. x² + 9x + 20 = 0

4. x² − 7x + 10 = 0

5. x² + 2x − 15 = 0

6. x² − x − 12 = 0

7. x² + 11x + 24 = 0

8. x² − 10x + 24 = 0

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Answers

1. x = −2, −3

2. x = 3, 5

3. x = −4, −5

4. x = 2, 5

5. x = 3, −5

6. x = 4, −3

7. x = −3, −8

8. x = 4, 6

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A quadratic equation is an equation whose highest exponent is two. Before attempting to solve it, identify whether it is already in standard form and determine the values of a, b, and c. When the equation can be factored, rewriting it as the product of two binomials often provides the quickest solution. The Zero Product Property then allows each factor to be solved separately, producing all possible solutions. As you continue practicing, you'll begin to recognize common factor patterns quickly, an essential skill for success on the Digital SAT Math section.