Algebra — Equations
GradesAlgebra IAlgebra — EquationsSolving Quadratic Equations

Solving Quadratic Equations

⚖️ Algebra — Equations
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Solving Quadratic Equations

Solve quadratics by factoring and the quadratic formula.

A quadratic equation has the standard form ax² + bx + c = 0. We solve quadratics by factoring (when factorable over the integers), completing the square, or applying the quadratic formula — which always works.

ax2+bx+c=0  ⟹  x=−b±b2−4ac2aax^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The discriminant b² − 4ac determines the nature of the solutions: if positive, two distinct real roots; if zero, exactly one real root (repeated); if negative, two complex conjugate roots.

Discriminant Key

b² − 4ac > 0 → Two distinct real rootsb² − 4ac = 0 → One repeated real rootb² − 4ac < 0 → Two complex (imaginary) roots
2x2−5x−3=0  ⟹  x=5±25+244=5±742x^2 - 5x - 3 = 0 \implies x = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm 7}{4}
x=5+74=3orx=5−74=−12x = \frac{5+7}{4} = 3 \quad \text{or} \quad x = \frac{5-7}{4} = -\frac{1}{2}
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Remember This!

Memorize the quadratic formula by the song "x equals negative b, plus or minus the square root of b squared minus 4ac, all over 2a." Computing the discriminant first tells you what type of answer to expect.

✏️ Try It!

Using the quadratic formula, what are the solutions to x² − 4x + 4 = 0?

Practice with CalcVerse
🌊Quadratic Formula Solver
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