This cheat sheet covers how to solve quadratic equations using the quadratic formula and how to interpret the discriminant. Students need it because many quadratics cannot be factored easily, but the quadratic formula works for every equation in standard form. It also helps students predict the number and type of solutions before solving.
The reference is organized to support quick checking during practice, homework, and test review.
The main form is , where . The quadratic formula is , and the discriminant is . If , there are two real solutions; if , there is one real repeated solution; if , there are two complex solutions.
Careful substitution, signs, and simplification are the keys to accurate answers.
Key Facts
- A quadratic equation in standard form is , where .
- The quadratic formula is .
- The discriminant is and it appears under the square root in the quadratic formula.
- If , the equation has two distinct real solutions.
- If , the equation has one real repeated solution, .
- If , the equation has two complex solutions because the square root of a negative number is not real.
- The axis of symmetry of the graph is .
- The solutions of are the -intercepts of the graph when the solutions are real.
Vocabulary
- Quadratic equation
- A polynomial equation of degree that can be written as with .
- Quadratic formula
- The formula used to solve any quadratic equation in standard form.
- Discriminant
- The expression that tells the number and type of solutions of a quadratic equation.
- Standard form
- The form , where the terms are arranged by descending powers of and one side equals .
- Repeated solution
- A single real solution that occurs when and the parabola touches the -axis at exactly one point.
- Complex solution
- A solution involving the imaginary unit , which occurs when the discriminant is negative.
Common Mistakes to Avoid
- Forgetting to set the equation equal to is wrong because , , and must come from standard form .
- Dropping the symbol is wrong because represents two possible solutions, and .
- Substituting incorrectly when is negative is wrong because means the opposite of , so if , then .
- Calculating without parentheses is wrong because signs matter, especially when or is negative.
- Dividing only the square root part by is wrong because the entire numerator must be divided by .
Practice Questions
- 1 Use the quadratic formula to solve .
- 2 Find the discriminant of and state the number of real solutions.
- 3 Solve and identify whether the solutions are real or complex.
- 4 Without solving, explain how the sign of tells whether the graph of crosses, touches, or does not meet the -axis.
Understanding Quadratic Formula & The Discriminant
The quadratic formula comes from completing the square, so it is not a rule that appeared from nowhere. Starting with a general quadratic, algebra can rearrange the terms until one side becomes a perfect square. Taking square roots then produces two possible values, one from adding and one from subtracting.
This is why the plus or minus sign must be treated as two separate solution paths. The expression under the square root measures whether that square root can be taken on the real number line. It is a compact result of the completing-the-square process.
The discriminant has a clear graph meaning. A parabola may cross the horizontal axis twice, touch it once at its turning point, or miss it entirely. These three pictures match the three possible discriminant cases.
When the parabola touches the axis, its turning point lies exactly on the axis. The two solutions have merged into the same number. The horizontal location of that turning point is negative b divided by two a.
This connection helps students check whether an answer makes visual sense. Two different real answers should give two crossing points. A repeated answer should place the vertex on the axis.
A reliable solving routine prevents most mistakes. First, move every term to one side so the equation equals zero. Next, identify the three coefficients, including their signs.
A negative coefficient should be written in parentheses when substituted into a calculation. Find the discriminant before doing the rest of the work. Then calculate the numerator for both the positive and negative square root paths, keep the denominator grouped, and simplify each result.
Fractions do not need to become decimals unless a question asks for approximations. Finally, substitute each answer into the original equation when possible. This catches sign errors that may be hidden in a long calculation.
Quadratics appear when a quantity changes with the square of another quantity. A ball thrown upward has a height that depends on time in a quadratic way. Its real solutions for height equal to zero can represent when it reaches the ground.
In geometry, a rectangle with a fixed perimeter can produce a quadratic relationship between its side lengths and area. In these settings, not every algebraic solution fits the situation. A negative time may be mathematically valid but physically meaningless.
Complex solutions can still be useful because they show that a model never reaches a chosen real value. Pay close attention to units, the meaning of the variable, and whether the final answers belong in the real world described by the problem.