Solving by Completing the Square (a ≠ 1): Full Solution
When the leading coefficient a ≠ 1, completing the square requires an extra first step — dividing every term by a — covered in California Reveal Math, Algebra 1 (Grade 9). Starting from ax²+bx+c=0, divide through by a so the x² coefficient becomes 1, giving x²+(b/a)x = -c/a. Then add (b/2a)² to both sides, factor the left side as a perfect square, and apply the Square Root Property. For example, solving 2x²-8x-10=0: divide by 2 to get x²-4x=5, add 4 to both sides to get (x-2)²=9, so x=5 or x=-1. Fractions often appear when a≠1, so careful arithmetic is essential. This method works on any quadratic and derives the quadratic formula.
Key Concepts
When $a \neq 1$, divide every term by $a$ first so the $x^2$ coefficient becomes 1, then complete the square normally:.
$$ax^2 + bx + c = 0 \;\longrightarrow\; x^2 + \frac{b}{a}x = \frac{c}{a}$$.
Common Questions
What is the first step when completing the square if a ≠ 1?
Divide every term in the equation by a so the coefficient of x² becomes 1. Then proceed with the standard completing-the-square steps.
How do you solve 2x²-8x-10=0 by completing the square?
Divide by 2: x²-4x=5. Add (−4/2)²=4 to both sides: x²-4x+4=9. Factor: (x-2)²=9. Square Root Property: x-2=±3, so x=5 or x=-1.
How do you compute the value to add when completing the square?
Take the coefficient of x after dividing by a, halve it, then square it: (b/2a)². Add this to both sides.
Can completing the square be used when a is not 1?
Yes, by dividing every term by a first. This transforms the equation into one where x² has coefficient 1, making the standard method applicable.
What is the Square Root Property used in completing the square?
If (x+k)² = d, then x+k = ±√d, so x = -k ± √d. This gives both solutions directly after factoring the perfect square.
What happens if a negative or fraction appears after dividing by a?
The process is the same — halve the resulting coefficient of x and square it. Work carefully with fractions to avoid arithmetic errors.
How does completing the square with a≠1 relate to the quadratic formula?
Applying this method to the general form ax²+bx+c=0 derives the quadratic formula x = (-b ± √(b²-4ac)) / 2a.