Look for the equation’s useful form
A quadratic equation has a nonzero squared term after like terms are combined. Three useful views are an expanded equation, a factored expression set equal to zero, and a squared expression isolated on one side. Before doing algebra, ask whether one of these views is already available. When a square is already isolated, expanding it can add work. When an equation factors cleanly, setting the product equal to zero can make the solutions visible. A solution is an input that satisfies the original equality.
Worked example with an isolated square
Solve (x + 4)² = 36 over the real numbers. First consider the whole expression inside the square. It can be 6 or −6, because either value squares to 36. Subtract 4 in both equations to get x = 2 or x = −10. Check each branch with the original expression. For 2, the inside is 6. For −10, the inside is −6. Both squares give 36. If the right side had been zero, there would be just one branch. A negative right side would give no real solution.
Use factors when a product equals zero
Consider x² + x − 12 = 0. Two integers that multiply to −12 and add to 1 are 4 and −3. That gives (x + 4)(x − 3) = 0. A product equals zero only when at least one factor is zero. The resulting values are −4 and 3. If the same product equals 12, setting the factors to zero would produce the wrong product. First move every term to one side so that the other side is zero, then choose a method that fits the resulting equation.
Connect an equation to a graph
For y = (x − 3)² − 5, the squared term is never negative. It reaches zero when x = 3. At that input the output is −5, so the vertex is (3, −5). An x-intercept has output zero. To find those intercepts, solve (x − 3)² = 5. The same expression answers two different questions, depending on whether you are looking for the minimum output or where the graph meets the horizontal axis.
Transfer check with a multiplier outside the square
Solve 2(x + 1)² = 98. Try to isolate the square before taking either branch. After dividing by 2, the squared expression equals 49, so x + 1 is 7 or −7. The solutions are 6 and −8. Plug each into the starting equation to check the factor of 2 as well as the sign. A single positive square root is useful for finding the size of each branch, but you still need both signs when the square equals a positive number.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original OpenExam explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
College Board ACCUPLACER Math test contentOpenStax Intermediate Algebra 2e