Solving equations by completing the square
Solving equations by completing the square is a Grade 7 algebra skill from Yoshiwara Intermediate Algebra that transforms a quadratic equation into the form (x+h)^2 = k and then solves using square roots. This technique works for any quadratic and helps students derive the quadratic formula.
Key Concepts
Property To solve a quadratic equation of the form $x^2+bx+c=0$: 1. Move the constant term to the other side: $x^2+bx = c$. 2. Complete the square on the left. Add $p^2 = (\frac{b}{2})^2$ to both sides of the equation: $x^2+bx+p^2 = c+p^2$. 3. Write the left side as a binomial squared: $(x+p)^2 = c+p^2$. 4. Use extraction of roots to find the solutions.
Examples To solve $x^2 6x 7 = 0$, first write $x^2 6x=7$. Add $(\frac{ 6}{2})^2=9$ to both sides: $x^2 6x+9=7+9$, so $(x 3)^2=16$. The solutions are $x=7$ and $x= 1$. To solve $x^2 4x 3 = 0$, first write $x^2 4x=3$. Add $(\frac{ 4}{2})^2=4$ to both sides: $x^2 4x+4=3+4$, so $(x 2)^2=7$. The solutions are $x = 2 \pm \sqrt{7}$. To solve $x^2+9x+20=0$, first write $x^2+9x= 20$. Add $(\frac{9}{2})^2=\frac{81}{4}$ to both sides: $x^2+9x+\frac{81}{4}= 20+\frac{81}{4}$, so $(x+\frac{9}{2})^2=\frac{1}{4}$. The solutions are $x= 4$ and $x= 5$.
Explanation This method transforms any quadratic equation into the form $(x+p)^2=q$. By forcing one side to be a perfect square, we can easily solve for $x$ by taking the square root of both sides, which is often simpler than factoring.
Common Questions
How do you solve a quadratic by completing the square?
Move the constant to the right side, add (b/2)^2 to both sides to complete the square on the left, factor the left side as a perfect square, then take the square root of both sides.
How do you solve x^2 + 4x - 5 = 0 by completing the square?
Rewrite as x^2 + 4x = 5. Add (4/2)^2 = 4 to both sides: x^2 + 4x + 4 = 9. Factor: (x+2)^2 = 9. Take roots: x+2 = ±3, giving x = 1 or x = -5.
Does completing the square always work?
Yes. Completing the square works for any quadratic, even ones that cannot be factored with integers.
What is the connection to the vertex of a parabola?
After completing the square, the equation is in vertex form y = a(x+h)^2 + k, directly revealing the vertex.