الخطط

حاسبة المعادلة التربيعية

تحل هذه الحاسبة ax² + bx + c = 0 بالقانون العام. أدخل a وb وc لتحصل على الجذور الدقيقة والمميّز وكل خطوة، مع التحقق من كل جذر.

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النتيجة

أدخل القيم واضغط احسب لرؤية النتيجة.

الصيغة

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The quadratic formula gives both roots of ax² + bx + c = 0 whenever a ≠ 0.

Δ=b2−4ac\Delta = b^2 - 4ac

The discriminant Δ tells you what kind of roots to expect: two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and a pair of complex conjugate roots if Δ < 0.

كيفية استخدام هذه الحاسبة

  1. Rearrange your equation so that one side is 0. For example, 2x² = 4x + 6 becomes 2x² − 4x − 6 = 0.
  2. Enter a, b and c. Use 0 for a missing term: x² − 9 = 0 has b = 0.
  3. Press Calculate. If you change an input afterwards, the result is marked as outdated until you calculate again.
  4. Read the exact roots first. Decimal approximations appear beneath them and are labelled as approximate.

أمثلة محلولة

Solve 2x² − 4x − 6 = 0

Here a = 2, b = −4 and c = −6.

The discriminant is Δ = (−4)² − 4 · 2 · (−6) = 16 + 48 = 64. It is positive, so there are two distinct real roots.

Apply the formula: x = (4 ± √64) / 4 = (4 ± 8) / 4.

So x = 12 / 4 = 3 or x = −4 / 4 = −1.

Check x = 3: 2 · 3² − 4 · 3 − 6 = 18 − 12 − 6 = 0.

Medium: roots with a square root

Solve the following quadratic equation: x² − 2x − 4 = 0. Here a = 1, b = −2 and c = −4.

Δ = (−2)² − 4 · 1 · (−4) = 4 + 16 = 20. It is positive but not a perfect square, so the two roots are irrational.

x = (2 ± √20) / 2 = (2 ± 2√5) / 2 = 1 ± √5, about 3.236068 and −1.236068.

Check: the roots add to 2 (= −b/a) and multiply to (1 + √5)(1 − √5) = 1 − 5 = −4 (= c/a).

Hard: complex roots

For x² + 2x + 5 = 0: Δ = 2² − 4 · 1 · 5 = 4 − 20 = −16. A negative discriminant means no real roots.

x = (−2 ± √−16) / 2 = (−2 ± 4i) / 2 = −1 ± 2i, where i² = −1.

Check: (−1 + 2i)² + 2(−1 + 2i) + 5 = (1 − 4i − 4) + (−2 + 4i) + 5 = 0.

أخطاء شائعة

  • Forgetting the minus in −b: for x² − 2x − 4 = 0, −b is +2, not −2.
  • Squaring a negative b wrongly: (−4)² is +16, so b² is never negative.
  • Dividing only part of the top by 2a: (2 ± √20) / 2 is 1 ± √5, not 2 ± √10.
  • Not rearranging first: 2x² = 4x + 6 must become 2x² − 4x − 6 = 0 before reading off a, b and c.

الأسئلة الشائعة

Is this a quadratic formula calculator with steps?

Yes. It shows the discriminant, the quadratic formula with your numbers substituted, the simplified roots and a check — every step, not just the answer.

Can this quadratic equation calculator solve any quadratic?

Any equation that can be written as ax² + bx + c = 0 with number coefficients — including fractions, decimals, irrational roots and complex roots.

What happens if a = 0?

Then the equation is not quadratic — it is the linear equation bx + c = 0. Sumzo tells you this and solves it as a linear equation (x = −c/b), or explains that there is no solution or that every number is a solution.

Why do some answers contain √ or i?

Sumzo keeps roots exact. When the discriminant is not a perfect square, the roots are irrational and are written with a square root, such as (−1 + √5)/2. When the discriminant is negative there are no real roots, and the roots are complex numbers written with i, where i² = −1. A rounded decimal is shown alongside each exact root.

What does “Verified” mean on this calculator?

After solving, Sumzo substitutes each exact root back into ax² + bx + c and confirms the result is exactly 0, and checks that the sum and product of the roots equal −b/a and c/a. These checks use exact arithmetic, not rounded decimals.

Can I enter fractions or decimals?

Yes. Enter values such as 0.5, −3 or 3/4. Use a period as the decimal point and leave out thousands separators.

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