Solving by completing the square
Solving by completing the square is a Grade 7 math skill from Yoshiwara Intermediate Algebra that transforms a quadratic equation into vertex form by adding a constant to both sides to create a perfect square trinomial. This method works for any quadratic and is the basis for deriving the quadratic formula.
Key Concepts
Property To Solve a Quadratic Equation by Completing the Square: 1. Write the equation in standard form. Divide both sides by the coefficient of the quadratic term, and subtract the constant term from both sides. 2. Complete the square on the left side: Multiply the coefficient of the first degree term by one half, then square the result. Add this value to both sides. 3. Write the left side of the equation as the square of a binomial. Simplify the right side. 4. Use extraction of roots to finish the solution.
Examples To solve $x^2 + 6x 7 = 0$, move the 7 over: $x^2 + 6x = 7$. Half of 6 is 3, and $3^2=9$. Add 9 to both sides: $x^2 + 6x + 9 = 16$. Factor the left side: $(x+3)^2 = 16$. So, $x+3 = \pm 4$, which gives $x=1$ and $x= 7$.
Solve $x^2 10x + 20 = 0$. Move 20 to get $x^2 10x = 20$. Half of 10 is 5, and $( 5)^2=25$. Add 25 to both sides: $x^2 10x + 25 = 5$. This becomes $(x 5)^2 = 5$, so the exact solutions are $x = 5 \pm \sqrt{5}$.
Common Questions
What is completing the square?
Completing the square rewrites ax^2 + bx + c = 0 by adding (b/2a)^2 to both sides to form a perfect square trinomial on one side, making it easy to solve.
How do you complete the square for x^2 + 6x = 7?
Take half of 6, which is 3, then square it to get 9. Add 9 to both sides: x^2 + 6x + 9 = 16. Factor: (x+3)^2 = 16, so x+3 = ±4, giving x = 1 or x = -7.
When is completing the square useful?
Completing the square is useful for solving quadratics that do not factor easily, converting to vertex form, and deriving the quadratic formula.
How is completing the square related to the quadratic formula?
The quadratic formula is derived by completing the square on the general quadratic ax^2 + bx + c = 0.