To complete the square and solve a quadratic equation, you need to manipulate the equation to have the constant on one side, eliminating the zero. Completing the square involves breaking a trinomial into two binomials, creating two possible solutions to the equation.
- Convert the quadratic equation into standard form
The standard form of a quadratic equation is ax^2 + bx + c.
- Move the constant, c, to the other side of the equation
If you begin with an equation of the form x^2 + 8x - 9 = 0, add 9 to both sides, leaving you with x^2 + 8x = 9.
- Halve the coefficient of x, square it, and add this to both sides
Doing this to our sample equation yields x^2 + 8x + 16 = 25.
- Rewrite the left side of the equation in simplified form
Here, the simplified form of x^2 + 8x + 16 is (x + 4)^2.
- Take the square root of both sides of the equation and solve for x
Remember that positive numbers have both a positive and a negative square root. In this case, because the right side of the equation is 25, it has square roots of 5 and -5. The equation is now x + 4 = +/- 5. Solving for x yields results of 1 and -9.
- Check your work
Plug both values of x into the original equation to check that they work. (1)^2 + 8(1) - 9 = 0, therefore 1 + 8 -9 = 0 holds true. (-9)^2 + 8(-9) - 9 =0, so 81 - 72 - 9 = 0 also holds true. Therefore, the two solutions in this example are correct.