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How do you solve a quadratic equation, and how do you choose between factoring, completing the square, and the quadratic formula?

Solve quadratic equations in one variable by factoring (zero-product property), completing the square, and the quadratic formula; recognize when a method is required by the problem; and interpret the solutions in a real-world context.

A NY Regents Algebra I answer on solving quadratics by factoring, completing the square, and the quadratic formula, when each is required, the zero-product property, and interpreting solutions in context.

Generated by Claude Opus 4.810 min answer

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  1. What this topic is asking
  2. Set to zero, then factor
  3. The quadratic formula
  4. Completing the square
  5. Choosing a method and interpreting in context
  6. Try this

What this topic is asking

The Regents Algebra I exam (the Reasoning with Equations and Inequalities, A-REI, cluster) expects you to solve a quadratic equation three ways, factoring, completing the square, and the quadratic formula, to recognize when a specific method is required, and to interpret the solutions in a context. Quadratics appear in Part I and almost always in a constructed-response question, frequently a projectile-motion or area model in Part III or Part IV.

Set to zero, then factor

The zero-product property is the engine of factoring: a product equals zero only when one factor is zero. So a quadratic must be set equal to zero before you factor.

x2+5x=24  ⇒  x2+5x−24=0  ⇒  (x+8)(x−3)=0  ⇒  x=−8 or x=3.x^2 + 5x = 24 \;\Rightarrow\; x^2 + 5x - 24 = 0 \;\Rightarrow\; (x + 8)(x - 3) = 0 \;\Rightarrow\; x = -8 \text{ or } x = 3.

The single most common error is factoring while one side is still nonzero; x2+5x=24x^2 + 5x = 24 does not mean x(x+5)=24x(x + 5) = 24 tells you anything useful, because 24 is not zero.

The quadratic formula

When a quadratic does not factor with integers, the quadratic formula always works.

The formula is on the Regents reference sheet, so the credit is for substituting correctly and simplifying, especially putting the radical in simplest form, 72=62\sqrt{72} = 6\sqrt{2}, when the question asks for exact answers.

Completing the square

Completing the square rewrites x2+bxx^2 + bx as a perfect square by adding (b2)2\left(\frac{b}{2}\right)^2. It is required when a question says "complete the square" and is the method behind both the vertex form and the quadratic formula itself.

Choosing a method and interpreting in context

Match the method to the question. If it factors, factoring is fastest. If the problem says "complete the square" or asks for the vertex, complete the square. If nothing factors, use the formula. In a contextual quadratic, the solutions are times, lengths, or counts, so a negative root is usually rejected. For a ball with height h(t)=−16t2+48th(t) = -16t^2 + 48t, setting h=0h = 0 gives t=0t = 0 (launch) and t=3t = 3 (landing); a negative time would be discarded.

A clarifying idea is that the three methods always agree, which gives you a check: a solution found by the formula should also satisfy the original equation when substituted back. Because the Regents awards method credit on the constructed-response parts, show every step, and where the prompt demands "simplest radical form", a rounded decimal will lose credit even if it is numerically close.

Try this

Q1. Solve (x−4)(x+7)=0(x - 4)(x + 7) = 0. [1 credit]

  • Cue. Zero-product property: x=4x = 4 or x=−7x = -7.

Q2. Use the discriminant to state the number of real roots of x2+2x+5=0x^2 + 2x + 5 = 0. [2 credits]

  • Cue. b2−4ac=4−20=−16<0b^2 - 4ac = 4 - 20 = -16 < 0, so no real roots.

Exam-style practice questions

Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Regents (style)2 marksPart I (multiple choice). What are the solutions to x2−6x=16x^2 - 6x = 16? (1) x=8x = 8 and x=−2x = -2 (2) x=−8x = -8 and x=2x = 2 (3) x=4x = 4 and x=4x = 4 (4) x=16x = 16 and x=0x = 0
Show worked answer →

The correct answer is (1).

First set the equation to zero: x2−6x−16=0x^2 - 6x - 16 = 0. Factor: find two numbers multiplying to −16-16 and adding to −6-6, namely −8-8 and 22, so (x−8)(x+2)=0(x - 8)(x + 2) = 0. By the zero-product property x=8x = 8 or x=−2x = -2. The most common mistake is factoring before moving 16 to the left, or reading the signs backward. Substituting x=8x = 8 gives 64−48=1664 - 48 = 16, confirming.

Regents (style)4 marksPart III (constructed response). Solve 2x2+4x−7=02x^2 + 4x - 7 = 0 using the quadratic formula. Express the answers in simplest radical form.
Show worked answer →

A 4-credit question: credit for the correct substitution, the simplification, and the radical form.

With a=2a = 2, b=4b = 4, c=−7c = -7: x=−4±42−4(2)(−7)2(2)=−4±16+564=−4±724x = \frac{-4 \pm \sqrt{4^2 - 4(2)(-7)}}{2(2)} = \frac{-4 \pm \sqrt{16 + 56}}{4} = \frac{-4 \pm \sqrt{72}}{4}. Simplify 72=62\sqrt{72} = 6\sqrt{2}: x=−4±624=−2±322x = \frac{-4 \pm 6\sqrt{2}}{4} = \frac{-2 \pm 3\sqrt{2}}{2}. Leaving 72\sqrt{72} unsimplified, or a sign error in −4ac-4ac, costs a credit. A decimal answer when "simplest radical form" is required also loses credit.

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