Calculator
Enter a quadratic expression in the form ax² + bx + c. Whole-number coefficients give the best rectangle factoring output.
Formula Used
ax2 + bx + c
ac = a × c
Find m and n where m + n = b and m × n = ac.
ax2 + mx + nx + c = grouped binomial factors.
x = (-b ± √(b2 - 4ac)) / 2a
How to Use This Calculator
Type the values of a, b, and c. Use the same signs shown in your expression. For example, enter -7 for a negative middle term. Then choose your variable. Press the factoring button. The result appears above the form and below the header.
Review the GCF first. Then check the ac product and middle split. The rectangle cells show how the expression is divided. Use the factored form for your final answer. Use the roots and graph to verify your work.
Use the CSV option for spreadsheets. Use the PDF option for homework records, tutoring notes, or classroom review.
Example Data Table
| Expression | ac Product | Middle Split | Factored Form |
|---|---|---|---|
| x² + 5x + 6 | 6 | 2x + 3x | (x + 2)(x + 3) |
| 2x² + 7x + 3 | 6 | 6x + x | (2x + 1)(x + 3) |
| 6x² - 5x - 6 | -36 | -9x + 4x | (3x + 2)(2x - 3) |
| 3x² + 12x + 12 | 12 after GCF | 2x + 2x | 3(x + 2)(x + 2) |
Generic Rectangle Factoring Guide
What This Method Does
The generic rectangle method changes a quadratic into a visual area model. It is useful when the leading coefficient is not one. It also helps students see why the middle term must be split. Each cell in the rectangle represents one part of the expression.
Why the ac Product Matters
The key step is finding the product of a and c. This product guides the search for two numbers. Their product must equal ac. Their sum must equal b. When those numbers are found, the middle term can be rewritten. Then the expression becomes easier to group.
How the Rectangle Is Built
The first cell contains the leading term. The last cell contains the constant term. The two remaining cells contain the split middle terms. The row and column factors then become the binomial factors. This structure makes the answer easier to check.
Using GCF First
A common factor should be removed before factoring. This makes the expression smaller. It also prevents missed answers. For example, 3x² + 12x + 12 has a common factor of 3. After removing it, the remaining trinomial is simpler.
Checking the Answer
The final factors should multiply back to the original expression. The calculator also displays roots, discriminant, vertex, and graph direction. These checks give extra confidence. If no integer rectangle factor exists, the expression may still factor with irrational or complex roots.
Best Use Cases
This calculator works well for algebra lessons, homework checks, test review, and tutoring. It shows the process, not only the answer. The export buttons help save the work for later study. The graph gives a quick visual review of the same expression.
FAQs
What is generic rectangle factoring?
It is a visual factoring method. It places quadratic terms inside a rectangle. The side lengths become the binomial factors.
What form should I enter?
Enter coefficients from ax² + bx + c. Use negative signs when needed. The calculator builds the expression from those values.
Why does the calculator use ac?
The ac product helps split the middle term. The split numbers must multiply to ac and add to b.
Does it handle a GCF?
Yes. The calculator checks for a greatest common factor first. Then it factors the remaining simpler trinomial.
Can it factor every quadratic?
It finds integer rectangle factors when they exist. Some quadratics need irrational, decimal, or complex factor forms.
Why are roots included?
Roots help verify the factorization. They also show where the graph crosses the horizontal axis when real roots exist.
What does the rectangle table show?
It shows the four area cells. These cells explain how the leading, middle, and constant terms are arranged.
Can I export my answer?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable result summary.