SAT — mathHard5 min read

The Discriminant and Nature of Solutions

Use the discriminant b²−4ac to determine the number and type of solutions for quadratic equations on the Digital SAT.

Tutor AI Team2026-02-25T01:23:11.204Z

The discriminant, $\Delta = b^2 - 4ac$ , is a powerful tool that tells you how many solutions a quadratic equation has without actually solving it. The Digital SAT tests the discriminant in questions about the nature of solutions, finding parameter values, and connecting algebra to graphical behaviour.

Core Concepts

The Discriminant

For the quadratic equation $ax^2 + bx + c = 0$ :

$\Delta = b^2 - 4ac$

Three Cases

Discriminant	Number of Solutions	Graph Behaviour
$\Delta > 0$	Two distinct real solutions	Parabola crosses x-axis twice
$\Delta = 0$	One repeated real solution	Parabola touches x-axis once
$\Delta < 0$	No real solutions	Parabola doesn't cross x-axis

Connection to the Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant is the expression under the square root. If it's negative, the square root is undefined over the reals, so there are no real solutions.

Finding Parameters Using the Discriminant

The SAT often asks: "For what value of $k$ does the equation have exactly one solution?"

Example: $x^2 + 6x + k = 0$ has exactly one solution. Find $k$ .

$\Delta = 36 - 4k = 0$ → $k = 9$

Example: $x^2 - 4x + k = 0$ has no real solutions. Find the range of $k$ .

$\Delta = 16 - 4k < 0$ → $k > 4$

Rational vs. Irrational Solutions

If $\Delta > 0$ :

$\Delta$ is a perfect square → two rational solutions (the equation factors nicely)
$\Delta$ is not a perfect square → two irrational solutions (involve $\sqrt{}$ )

Strategy Tips

Tip 1: Memorise the Three Cases

$\Delta > 0$ : two solutions. $\Delta = 0$ : one solution. $\Delta < 0$ : no real solutions.

Tip 2: Set $\Delta = 0$ for "Exactly One Solution"

This is the most common SAT question type involving the discriminant.

Tip 3: Correctly Identify $a$ , $b$ , $c$

Make sure the equation is in standard form $ax^2 + bx + c = 0$ before computing the discriminant. Watch for negative $b$ values.

Tip 4: Connect to Graphs

Does $3x^2 + 2x + 1 = 0$ have real solutions?

$\Delta = 4 - 12 = -8 < 0$ → No real solutions.

Solution

Practice Problems

Problem 1

How many solutions does $x^2 + 4x + 4 = 0$ have?

Problem 2

For what value of $k$ does $x^2 - 8x + k = 0$ have exactly one real solution?

Problem 3

For what values of $m$ does $x^2 + mx + 9 = 0$ have no real solutions?

Problem 4

The equation $x^2 - 6x + c = 0$ has two distinct real solutions. What is the range of $c$ ?

Problem 5

Is $2x^2 - 7x + 3 = 0$ factorable over the integers? (Check if $\Delta$ is a perfect square.)

Problem 6

A parabola $y = x^2 + bx + 10$ doesn't cross the x-axis. What are the possible values of $b$ ?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Forgetting the $-4ac$ term. The discriminant is $b^2 - 4ac$ , not $b^2 - 4c$ or $b^2 + 4ac$ .
Wrong sign for $b$ . If the equation is $x^2 - 6x + 5 = 0$ , then $b = -6$ and $b^2 = 36$ .
Not putting the equation in standard form first. Rearrange to $ax^2 + bx + c = 0$ before identifying $a$ , $b$ , $c$ .
Confusing "one solution" with "no solution." $\Delta = 0$ → one solution (double root). $\Delta < 0$ → no real solutions.
Arithmetic errors. Compute $b^2$ and $4ac$ carefully, especially with negative values.

Frequently Asked Questions

Can the discriminant be used for non-quadratic equations?

No — the discriminant $b^2 - 4ac$ is specific to quadratic equations.

What does a repeated root mean graphically?

The parabola is tangent to the x-axis (touches it at exactly one point).

Does the discriminant tell me the actual solutions?

No — it only tells you how many and what type. Use the quadratic formula for the actual values.

How often is this on the SAT?

Very common. Expect 1–2 discriminant-related questions per test.

What if $a$, $b$, or $c$ contains a variable?

Treat the discriminant as an expression in that variable and set up the appropriate inequality.

Key Takeaways

✓
$\Delta = b^2 - 4ac$ determines the number of real solutions.
✓
$\Delta > 0$ : two solutions. $\Delta = 0$ : one solution. $\Delta < 0$ : none.
✓
Set $\Delta = 0$ to find the parameter value giving exactly one solution.
✓
Set $\Delta > 0$ or $\Delta < 0$ for two solutions or no solutions respectively.
✓
Perfect square discriminant means the quadratic factors over the integers.
✓
Always identify $a$ , $b$ , $c$ correctly from standard form.

Tags:

#sat#math#advanced-math#quadratics#discriminant

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Core Concepts

The Discriminant

Three Cases

Connection to the Quadratic Formula

Finding Parameters Using the Discriminant

Rational vs. Irrational Solutions

Strategy Tips

Tip 1: Memorise the Three Cases

Tip 2: Set Δ=0\Delta = 0Δ=0 for "Exactly One Solution"

Tip 3: Correctly Identify aaa, bbb, ccc

Tip 4: Connect to Graphs

Tip 5: Perfect Square Discriminant = Factorable

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: SAT-Style

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

Ready to Ace Your SAT math?

Tip 2: Set $\Delta = 0$ for "Exactly One Solution"

Tip 3: Correctly Identify $a$ , $b$ , $c$