Quadratic Formula Calculator
Enter coefficients for ax² + bx + c = 0:
Enter coefficients for ax² + bx + c = 0:
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The quadratic formula is a fundamental tool in algebra used to solve quadratic equations. A quadratic equation is a second-degree polynomial equation of the form:
Here, \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents the variable. The quadratic formula provides the solutions (or roots) of this equation, which are the values of \( x \) that satisfy the equation.
The quadratic formula is given by:
This formula calculates the roots of the quadratic equation by taking into account the coefficients \( a \), \( b \), and \( c \). The term under the square root, \( b^2 - 4ac \), is called the discriminant. The discriminant determines the nature of the roots:
Let’s break down the steps to solve a quadratic equation using the quadratic formula:
Solve the quadratic equation \( 2x^2 - 4x - 6 = 0 \).
Step 1: Identify the coefficients: \[ a = 2, \quad b = -4, \quad c = -6 \]
Step 2: Calculate the discriminant: \[ D = b^2 - 4ac = (-4)^2 - 4(2)(-6) = 16 + 48 = 64 \]
Step 3: Since \( D > 0 \), there are two distinct real roots.
Step 4: Apply the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4} \] This gives two solutions: \[ x_1 = \frac{4 + 8}{4} = 3 \] \[ x_2 = \frac{4 - 8}{4} = -1 \]
Solution: The roots are \( x = 3 \) and \( x = -1 \).
If the discriminant is negative, the quadratic formula will yield complex roots. For example, if \( D = -9 \), the roots will be: \[ x = \frac{-b \pm \sqrt{-9}}{2a} = \frac{-b \pm 3i}{2a} \] where \( i \) is the imaginary unit (\( i^2 = -1 \)).
The quadratic formula is a powerful and straightforward method for solving quadratic equations. By understanding the discriminant and following the steps outlined above, you can easily determine the roots of any quadratic equation. Whether the roots are real or complex, the quadratic formula provides a clear and systematic way to find them.