Our intuitive quadratic equation calculator is the perfect tool for students, educators, and professionals alike who need to swiftly and accurately solve quadratic equations.
By simply inputting the coefficients a, b, and c, the calculator automatically applies the quadratic formula to determine the roots, making complex calculations a breeze.
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It not only gives you the final answers but also shows you the detailed steps involved in the solution process, enhancing your understanding and reinforcing key mathematical concepts.
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The quadratic equation is:
\[ ax^2 + bx + c = 0 \]
To solve for \( x \), we use the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
We can help you calculate an equation of the form "ax2 + bx + c = 0"
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In algebra, the quadratic formula is the solution to the quadratic equation. The quadratic equation is a second-order polynomial equation in a single variable \( x \) with a non-zero coefficient for \( x^2 \). The quadratic formula provides the solution(s) to the equation:
\[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are coefficients of the equation, and \( a \neq 0 \).
Let's solve the quadratic equation \( 2x^2 + 4x - 6 = 0 \).
Here, \( a = 2 \), \( b = 4 \), and \( c = -6 \). Using the quadratic formula:
\[ x = \frac{{-4 \pm \sqrt{{4^2 - 4 \cdot 2 \cdot (-6)}}}}{{2 \cdot 2}} \]
Simplifying inside the square root:
\[ x = \frac{{-4 \pm \sqrt{{16 + 48}}}}{{4}} = \frac{{-4 \pm \sqrt{{64}}}}{{4}} \]
Simplifying further:
\[ x = \frac{{-4 \pm 8}}{{4}} \]
So, we have two solutions:
\[ x_1 = \frac{{-4 + 8}}{{4}} = 1 \quad \text{and} \quad x_2 = \frac{{-4 - 8}}{{4}} = -3 \]
Therefore, the roots of the equation \( 2x^2 + 4x - 6 = 0 \) are \( x_1 = 1 \) and \( x_2 = -3 \).
Let's solve the quadratic equation \( x^2 + 2x + 5 = 0 \).
Here, \( a = 1 \), \( b = 2 \), and \( c = 5 \). Using the quadratic formula:
\[ x = \frac{{-2 \pm \sqrt{{2^2 - 4 \cdot 1 \cdot 5}}}}{{2 \cdot 1}} \]
Simplifying inside the square root:
\[ x = \frac{{-2 \pm \sqrt{{4 - 20}}}}{{2}} = \frac{{-2 \pm \sqrt{{-16}}}}{{2}} \]
Since we have a negative number under the square root, we get complex roots:
\[ x = \frac{{-2 \pm 4i}}{{2}} = -1 \pm 2i \]
Therefore, the roots of the equation \( x^2 + 2x + 5 = 0 \) are \( x_1 = -1 + 2i \) and \( x_2 = -1 - 2i \).