9.2 Completing the Square and the Quadratic Formula

9.2 Completing the Square and the Quadratic Formula

Quick Answer: When a quadratic will not factor cleanly, the Florida Algebra 1 EOC (B.E.S.T.) expects two more methods under MA.912.AR.3.5 and MA.912.AR.3.6: completing the square (which also converts standard form to vertex form) and the quadratic formula. The discriminant b^2 - 4ac tells you the nature of the roots before you solve, and any method that involves squaring can introduce extraneous solutions you must check.

Completing the Square (MA.912.AR.3.5)

Completing the square rewrites ax^2 + bx + c as a(x + h)^2 + k, which is vertex form. The same procedure solves any quadratic, even one that does not factor over the integers.

Procedure (a = 1)

1. Move the constant to the other side: x^2 + bx = -c.
2. Take half of b, square it, add to both sides.
3. Write the left side as a perfect square.
4. Take square roots and solve for x.

Worked example. Solve x^2 + 6x - 7 = 0. Add 7: x^2 + 6x = 7. Half of 6 is 3; add 9 to both sides: x^2 + 6x + 9 = 16. Write (x + 3)^2 = 16. Take square roots: x + 3 = +/- 4. Solve: x = 1 or x = -7.

When a != 1

Divide every term by a first. For 2x^2 - 8x + 4 = 0, divide by 2 to get x^2 - 4x + 2 = 0, then complete the square. Half of -4 is -2; add 4 to both sides: (x - 2)^2 = 2, so x = 2 +/- sqrt(2). Dividing by a is the step students forget most often.

Converting to Vertex Form

The same steps convert standard form to vertex form. For f(x) = x^2 + 6x + 4: add 9 to both sides of x^2 + 6x = f(x) - 4 to get (x + 3)^2 = f(x) + 5, so f(x) = (x + 3)^2 - 5. Vertex: (-3, -5). The EOC uses this form to ask about transformations.

The Quadratic Formula (MA.912.AR.3.6)

For ax^2 + bx + c = 0 with a != 0:

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

The formula always works, including when factoring and completing the square are ugly. Memorize it exactly; a common EOC trap puts -b in the numerator with the wrong sign, or swaps 2a for a^2.

Worked example. Solve 2x^2 - 5x - 3 = 0.

1. Identify a = 2, b = -5, c = -3.
2. Discriminant: (-5)^2 - 4(2)(-3) = 25 + 24 = 49.
3. sqrt(49) = 7.
4. x = (5 +/- 7) / 4.
5. x = 12/4 = 3, or x = -2/4 = -1/2.
6. Solution set: {-1/2, 3}.

Discriminant and Nature of Roots

The expression under the radical, b^2 - 4ac, is the discriminant. It tells you the nature of the roots before you solve:

• b^2 - 4ac > 0, perfect square: two distinct rational roots; parabola crosses x-axis twice; factoring works.
• b^2 - 4ac > 0, not a perfect square: two distinct irrational roots; parabola crosses x-axis twice; formula required.
• b^2 - 4ac = 0: one real repeated root; parabola touches x-axis at the vertex (tangent).
• b^2 - 4ac < 0: no real roots; parabola does not cross the x-axis.

Reading the discriminant first saves time: if it is negative, do not try to factor; the answer is "no real roots." If it is a perfect square and a = 1, factoring is usually faster than the formula.

Extraneous Solutions

An extraneous solution is a value that emerges from the algebra but does not satisfy the original equation. On the Algebra 1 EOC, extraneous roots appear most often when you square both sides of an equation that contains a radical or when the quadratic came from multiplying both sides by a variable expression.

Worked example. Solve sqrt(x + 4) = x - 2. Square both sides: x + 4 = (x - 2)^2 = x^2 - 4x + 4. Set to zero: x^2 - 5x = 0, so x(x - 5) = 0. Candidates: x = 0 or x = 5. Check in the original equation: x = 0 gives sqrt(4) = -2, which is false (extraneous); x = 5 gives sqrt(9) = 3, which is true. Only solution: x = 5.

The EOC almost always includes the extraneous candidate as a distractor. If you do not check both roots against the original equation, you will pick the wrong answer.

Common EOC Mistakes on This Benchmark

• Sign error on -b. For 2x^2 - 5x - 3 = 0, -b = 5, not -5. The numerator starts with the opposite sign of b.
• Forgetting to divide by 2a. Writing x = 5 +/- 7 instead of (5 +/- 7)/4 gives wrong values.
• Assuming a = 1. If the leading coefficient is not 1, handle a correctly in both the formula and completing the square.
• Reporting complex roots. The Algebra 1 course stops at real roots; "no real roots" is the expected response when the discriminant is negative.
• Skipping the check after squaring. Every candidate from a squared equation must be verified in the original equation.

Choosing Among All Four Methods

The EOC method-choice logic: (1) b = 0? Square-root property. (2) Factorable? Factoring. (3) Need vertex form? Completing the square. (4) Otherwise? Quadratic formula. The discriminant is the fastest pre-check: negative means "no real roots," a perfect square means factoring works, and any other positive value means the formula gives irrational roots in simplified form.