Solution : Solve the Quadratic Equation by Completing the Square

Solve the quadratic equation by Completing the Square

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Solution

In this page we demonstrate how to solve a quadratic equation using the completing the square method.
This classical technique rewrites a general form ax² + bx + c = 0 as a perfect square, which simplifies solving for x, even when roots are irrational or complex. :contentReference[oaicite:1]{index=1}
The method also reveals important properties — for example, the vertex of the parabola defined by the quadratic — and forms the basis for the general quadratic formula. :contentReference[oaicite:2]{index=2}
Below, you will see a clear, step-by-step worked example that leads to the full solutions of the equation.

\[\begin{aligned} &3 x^2+4 x-15=0 \\\\ &x^2+\frac{4}{3} x-5=0 \\\\ &x^2+2\left(\frac{2}{3}\right) x+\left(\frac{2}{3}\right)^2-\left(\frac{2}{3}\right)^2-5=0 \\\\ &\left(x+\frac{2}{3}\right)^2=5+\frac{4}{9} \\\\ &\left(x+\frac{2}{3}\right)^2=\frac{45}{9}+\frac{4}{9} \\\\ &\left(x+\frac{2}{3}\right)^2=\frac{49}{9} \\\\ &x+\frac{2}{3}=\pm \frac{7}{3} \\\\ &x=\pm \frac{7}{3}-\frac{2}{3} \\\\ &x=\frac{7}{3}-\frac{2}{3} \text { or } x=\frac{-7}{3}-\frac{2}{3} \\\\ &x=\frac{5}{3} \text { or } x=-3 \end{aligned}\]

\[\text {Thus, the solution set}=\left\{-3, \frac{5}{3}\right\}\]

By completing the square, we transformed the quadratic equation into a perfect square trinomial and solved precisely for x.
This method is versatile: it works for any quadratic (even when roots are irrational or complex), and reveals more than just the roots — it also gives insight into the parabola’s vertex and its symmetry.
Remember that the key steps are: normalizing the coefficient of x² to 1 (if needed), moving the constant, adding \((b/2a)^2\) appropriately, then solving the resulting equation.
For more algebraic problems and solutions, feel free to browse our Solved Exercises section.

Home -> Solved problems -> Solve the quadratic equation by Completing the Square

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