A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0, and it can have two real solutions, one, or none. Algebra I gives you four tools for solving it. They all give the same answers, so the skill is choosing the quickest one and carrying it out cleanly. Each tool also reveals something different about the expression, which is the point of A-SSE.3.
Method 1: Factoring
If the expression factors, set each factor equal to zero. This works because a product is zero only when one of its factors is zero.
Worked example. Solve x² + 5x + 6 = 0. Look for two numbers that multiply to 6 and add to 5: 2 and 3. So (x + 2)(x + 3) = 0, which gives x = −2 or x = −3. Check x = −2: 4 − 10 + 6 = 0. Check x = −3: 9 − 15 + 6 = 0.
Factoring a quadratic shows the zeros of the function it defines. That is the first part of A-SSE.3.
Method 2: Taking square roots
When the equation has a squared quantity and no separate x term, isolate the square and take both roots.
Worked example. Solve (x − 4)² = 25. Then x − 4 = 5 or x − 4 = −5, so x = 9 or x = −1. Check x = −1: (−5)² = 25.
Another: 3x² = 48 gives x² = 16, so x = 4 or x = −4. Always keep both signs; losing the negative root is the most common slip here.
Method 3: Completing the square
Completing the square rewrites x² + bx + c so a perfect square appears. Add the square of half the x coefficient to both sides.
Worked example. Solve x² + 6x − 7 = 0.
- Move the constant: x² + 6x = 7.
- Half of 6 is 3, and 3² = 9. Add 9 to both sides: x² + 6x + 9 = 16.
- Factor the left side: (x + 3)² = 16.
- Take roots: x + 3 = 4 or x + 3 = −4, so x = 1 or x = −7.
Check x = 1: 1 + 6 − 7 = 0. Check x = −7: 49 − 42 − 7 = 0.
This method also shows a function's extreme value, the second part of A-SSE.3. Write x² − 6x + 5 as (x − 3)² − 4. Since a square is never negative, the smallest value is −4, reached at x = 3. Check: 9 − 18 + 5 = −4.
When the roots are not whole numbers, completing the square still works. Solve x² + 8x + 3 = 0: x² + 8x = −3, add 16 to get (x + 4)² = 13, so x = −4 + √13 or x = −4 − √13. Check: (x + 4)² = 13, and 16 − 13 = 3, matching the constant.
Method 4: The quadratic formula
For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. It works on every quadratic, which makes it the dependable fallback.
Worked example. Solve 2x² + 3x − 5 = 0, where a = 2, b = 3, c = −5. The part under the root is b² − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49. So x = (−3 ± 7) ÷ 4, which gives x = 4 ÷ 4 = 1 or x = −10 ÷ 4 = −2.5. Check x = −2.5: 2(6.25) − 7.5 − 5 = 0.
Worked example with irrational roots. Solve x² − 4x + 1 = 0. Here b² − 4ac = 16 − 4 = 12, and √12 = 2√3. So x = (4 ± 2√3) ÷ 2 = 2 ± √3, about 3.73 or 0.27.
What the discriminant tells you
The expression b² − 4ac decides how many real solutions exist. If it is positive there are two, if it is zero there is one, and if it is negative there are none. For x² + 2x + 5 = 0 it equals 4 − 20 = −16, so there are no real solutions and the graph never touches the x-axis.
The discriminant also predicts which methods will work well. If b² − 4ac is a perfect square (such as 49 above), the quadratic factors over the integers, so factoring will succeed. If it is positive but not a perfect square (such as 12), the roots are irrational and factoring over the integers will not work; use completing the square or the formula.
Choosing a method
- Squared quantity with no extra x term, such as (x + 2)² = 9: take square roots.
- Small whole-number coefficients that factor quickly: factor.
- Leading coefficient 1 and an even middle coefficient, or you need the vertex form: complete the square.
- Anything else, or when factoring stalls after a short try: quadratic formula.
Common mistakes
- Not setting the equation equal to zero before factoring. From x² + x = 6 you cannot read off x = 6 or x + 1 = 6; first write x² + x − 6 = 0.
- Dropping the ± sign when taking roots.
- Adding the half-coefficient square to one side only when completing the square.
- Sign errors in the formula, especially −b when b is already negative, and the −4ac term when c is negative.
Practice Algebra I questions with an explanation for every answer, or read the guide to California Algebra I practice.
Source: the California Common Core State Standards for Mathematics, A-REI.4, A-SSE.3.