Enter a Quadratic Expression
Use the form for expressions written as ax² + bx + c. Fractions like 3/4 are accepted.
Formula Used
Standard form: ax2 + bx + c
Discriminant: D = b2 - 4ac
Roots: r1 = (-b + √D) / 2a and r2 = (-b - √D) / 2a
Factored form: a(x - r1)(x - r2)
The calculator first identifies the discriminant. It then uses the roots to build the factor pattern. Positive, zero, and negative discriminants are handled separately.
How to Use This Calculator
Enter the values of a, b, and c from the expression ax² + bx + c. Keep every sign exactly as written. Choose a variable, precision, and answer style. Use the optional check value when you want to test a point. Press the calculate button. The result will appear above the form.
Quadratic Form to Factored Form Guide
A quadratic in standard form is written as ax² + bx + c. Factored form shows the same expression as a product of simpler linear factors. This view makes the roots easier to read. It also helps when solving equations, drawing graphs, and checking intercepts. The calculator accepts any real coefficients and returns a clear factor pattern when one exists.
Why Factored Form Matters
Factored form is useful because each factor points to a zero. If the expression is a(x - r₁)(x - r₂), the graph crosses or touches the x-axis at r₁ and r₂. These values are the roots. A repeated root means the parabola touches the axis once. A negative discriminant means real linear factors are not available.
Formula Used
The main formula comes from the quadratic equation. For ax² + bx + c, the discriminant is D = b² - 4ac. The roots are r₁ = (-b + √D) / 2a and r₂ = (-b - √D) / 2a. The factored form is a(x - r₁)(x - r₂).
Input Checks
Always confirm that a is not zero. When a is zero, the expression is linear, not quadratic. Signs also matter. A positive c and negative b can create two positive roots. A positive c and positive b can create two negative roots. Decimal inputs may give rounded roots, so use enough precision.
Exact And Decimal Results
Exact results are best for classroom work because they keep radicals and fractions visible. Decimal results are helpful when coefficients are measured values. This calculator can show both views. It also reports the discriminant, vertex, axis of symmetry, y-intercept, and a sample expansion check. These details help catch common errors.
How To Use This Calculator
Enter the coefficients a, b, and c from the quadratic expression. Choose a variable symbol if needed. Select the precision for decimal roots. Decide whether to show detailed steps and extra graph facts. Press the calculate button. The answer appears above the form, so the inputs remain easy to adjust.
Reading The Answer
When the roots are real, the result shows a product of factors. For example, x² - 5x + 6 becomes (x - 2)(x - 3). When the discriminant is zero, the answer uses a squared factor. When the discriminant is negative, the page explains that the expression needs complex factors.
Study Tips
Use the expansion check to verify the answer. Multiply the factors back together. The result should return the original quadratic. If it does not match, check the signs first. Then check the leading coefficient. Factoring becomes easier when roots, coefficients, and graph behavior are studied together.
Common Mistakes
Many wrong factors come from changing signs soon. Remember that x - r is used when the root is r. Another mistake is dropping the leading coefficient. Keep a outside the factors unless the factors are rebuilt with integer coefficients. Complex roots are valid, but they are not real x-intercepts on the graph.
FAQs
What is quadratic form?
Quadratic form usually means standard form, ax² + bx + c. The value of a cannot be zero. This calculator converts that form into a product of factors when the roots allow it.
What is factored form?
Factored form writes a quadratic as a product, such as a(x - r₁)(x - r₂). The values r₁ and r₂ are roots. They show where the graph crosses or touches the x-axis.
Which formula does the calculator use?
It uses D = b² - 4ac to find the discriminant. Then it uses r = (-b ± √D) / 2a. The roots are placed into a(x - r₁)(x - r₂).
Can it factor quadratics with decimal coefficients?
Yes. Enter decimal coefficients directly. The calculator can display rounded decimal roots. Exact symbolic output may be less clean when coefficients are decimals or measured values.
Can I enter fractions?
Yes. Simple fractions like 3/4 and -5/2 are accepted. Do not include mixed numbers. Convert mixed numbers into improper fractions before entering them.
What happens when the discriminant is zero?
A zero discriminant gives one repeated root. The factored form uses a squared factor, such as a(x - r)². The parabola touches the x-axis once.
What happens when the discriminant is negative?
A negative discriminant means there are no real roots. The calculator explains that real linear factors are not available. You can enable complex factors to see conjugate roots.
Why is coefficient a important?
The coefficient a controls the leading scale and direction of the parabola. It must stay in the factored form unless the factors are rewritten with adjusted integer coefficients.
How do I check the factored answer?
Multiply the factors back together. The expanded result should match ax² + bx + c. You can also enter a check value to compare an evaluated output.
Does factored form show x-intercepts?
Yes, when the roots are real. Each factor gives one x-intercept. Repeated roots show a single touching point, not two crossing points.
Can this help with graphing?
Yes. It reports roots, vertex, axis of symmetry, and y-intercept when details are enabled. These values help sketch the parabola accurately.