Solve quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. Get detailed solutions including discriminant analysis, roots, vertex, and axis of symmetry.
Coefficient of x²
Coefficient of x
Constant term
The quadratic formula is a powerful tool for solving any quadratic equation of the form ax² + bx + c = 0. It provides a systematic method to find the roots (solutions) of the equation, regardless of whether they are real or complex numbers.
x = [-b ± √(b² - 4ac)] / 2a
a = coefficient of x² (quadratic term)
b = coefficient of x (linear term)
c = constant term
± = plus-minus symbol (gives two solutions)
The discriminant (b² - 4ac) determines the nature and number of solutions:
Two distinct real roots
Parabola crosses x-axis twice
One real root (repeated)
Parabola touches x-axis once
Two complex roots
Parabola doesn't cross x-axis
The vertex is the highest or lowest point on the parabola. It's located at:
x = -b / 2a
y = f(x) at vertex x
The vertical line that divides the parabola into two mirror images:
x = -b / 2a
Projectile motion, calculating maximum height and range of objects
Designing parabolic structures like bridges and satellite dishes
Profit optimization, finding break-even points and maximum revenue
Creating curved paths and animations in games and simulations
If a > 0, the parabola opens upward (U-shaped) with a minimum at the vertex. If a < 0, it opens downward (∩-shaped) with a maximum at the vertex.
When the discriminant is negative, the roots are complex numbers in the form a + bi, where i is the imaginary unit (√-1). These roots always come in conjugate pairs.
For any quadratic equation: Sum of roots = -b/a, Product of roots = c/a. These relationships are useful for checking solutions.
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