Solving Quadratic Equations by Completing the Square

Every standard quadratic equation is represented in the form $ax^2 + bx + c = 0$. To solve it by completing the square, the calculator performs a systematic sequence of algebraic transformations. First, it divides the entire equation by the leading coefficient $a$ to normalize the $x^2$ term, yielding $x^2 + b_1x + c_1 = 0$, where $b_1 = b/a$ and $c_1 = c/a$. Next, it isolates the constant term on the right side and focuses on the linear coefficient $b_1$. By taking half of $b_1$ (denoted as $b_2 = b_1/2$) and squaring it ($b_2^2$), the calculator determines the exact value needed to turn the left side into a perfect square trinomial. This value is added to both sides of the equation. Finally, the equation takes the form $(x + b_2)^2 = d_3$, allowing the tool to take the square root of both sides and isolate $x$ to find the final roots.

Let us solve the quadratic equation $2x^2 + 8x - 10 = 0$ using our methodology. First, identify the coefficients: $a = 2$, $b = 8$, and $c = -10$. Step one normalizes the equation by dividing all terms by $a$ ($2$), giving us $x^2 + 4x - 5 = 0$. Here, $b_1 = 4$ and $c_1 = -5$. Step two shifts the constant term to the right side, resulting in $x^2 + 4x = 5$. Step three takes half of the linear coefficient $b_1$ ($4 / 2 = 2$) and squares it to get $4$. We add this value to both sides: $x^2 + 4x + 4 = 5 + 4$, which simplifies to the binomial square $(x + 2)^2 = 9$. Taking the square root of both sides yields $x + 2 = \pm3$. Solving for $x$ gives our two distinct roots: $x_1 = 3 - 2 = 1$ and $x_2 = -3 - 2 = -5$.

When working with quadratic equations manually or verifying calculator results, keep these pointers in mind. Always ensure your initial equation is written in standard form ($ax^2 + bx + c = 0$) before plugging in numbers. Watch out for negative signs, especially when dividing negative constants or dealing with non-real complex numbers when the discriminant ($d_3$) is negative. Using precision settings appropriately will help you manage repeating decimals effectively.