Math · College algebra · Worked example
Find both zeros of a quadratic
Solve x² − 5x + 6 = 0, then connect the answer to a graph.
Identify what we are solving
We want all real values of x that make the expression zero. Because the coefficient of x² is 1, we can start by looking for two numbers with product 6 and sum −5.
Choose and check the factors
The numbers −2 and −3 work. Their product is positive, and their sum is −5. Multiply back once to check the middle term before using the factorization.
Rewrite the equation
Replace the polynomial with its factored form. We have changed how the equation is written, but we have kept the same solutions.
Keep both possibilities
For the product to be zero, either x − 2 is zero or x − 3 is zero. Solve each small equation. Neither possibility should be dropped.
Why not divide by x − 2?
Dividing by x − 2 would assume x is not 2. That would discard one of the solutions we are trying to find. Setting each factor equal to zero keeps both possibilities.
Substitute each answer into the original equation
Check x = 2 and x = 3 in the original expression. Both give zero, so both are valid solutions. These substitutions also give a quick independent check of our signs.
Predict the graph before opening it
The curve should cross the horizontal axis at 2 and 3. At x = 2.5, its height is −0.25, so it should dip below the axis between the roots. Open the graph to compare those predictions with the curve.
Result
The solutions are x = 2 and x = 3.
Your turn
Solve x² − 7x + 12 = 0. Check both solutions in the original equation and predict where the graph is negative.
Show the answer and explanation
The solutions are x = 3 and x = 4. The graph is negative for 3 < x < 4.
The numbers −3 and −4 have product 12 and sum −7, so the factors are (x − 3)(x − 4). Each factor gives one zero. Between the zeros the factors have opposite signs.
Keep exploring
Use the workspace to explore a changed input. Return to the concept if you want to revisit the reasoning.
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.