Quadratic Formula on the SAT
The Reliable Backup Plan
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Quadratic Formula
Not every quadratic factors cleanly, and the SAT knows that. When factoring fails, the quadratic formula is the guaranteed method that always works.
The formula looks intimidating because it is long, but the steps are systematic. This lesson shows how to identify $a$, $b$, and $c$ correctly, substitute carefully, and simplify without losing the $\pm$ sign.
A Simple Definition Unlocks Quadratic Formula
The quadratic formula applies to any equation in the form $ax^2 + bx + c = 0$. Once you know $a$, $b$, and $c$, you plug them into $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
The expression under the square root, called the discriminant, tells you how many real solutions exist. A positive discriminant gives two solutions, zero gives one, and a negative discriminant gives none in the real number system.
Work Through Quadratic Formula Step by Step
Solve $x^2 - 4x - 1 = 0$ using the quadratic formula.
Use a careful substitution into the quadratic formula and simplify step by step.
Identify the coefficient $a$ from the quadratic.
Identify the coefficient $b$ from the quadratic.
Identify the constant term $c$ from the quadratic.
Substitute into the quadratic formula
Simplify the discriminant to see the number of solutions.
Finish the calculation to isolate the solution.
Simplify the radical so the solution is in simplest form.
Reduce the fraction to its simplest form.
Use Desmos to Check Quadratic Formula
Solve $x^2 - 4x - 1 = 0$ using the quadratic formula.
Desmos can find roots by graphing the quadratic and reading the $x$ intercepts. This is a fast check when the formula gives complicated radicals.
y = x^2 - 4x - 1
Algebra is required when the question asks for an exact form like $2 \pm \sqrt{5}$. Desmos is faster when a decimal approximation is enough.
Desmos is faster for approximate roots or for checking your result. Algebra is faster when the question requires exact radical answers.
Expert move: Graph the quadratic and click the $x$-intercepts and vertex to read solutions and key features; the graph makes it clear when there are 0, 1, or 2 real roots.
Precision check: Use Desmos for decimal answers or verification, but convert to a fraction if the choices are exact and apply any context restrictions.
- Desmos features used: graphing, $x$ intercepts.
- Common mistake: forgetting the $\pm$ in the formula.
Practice Quadratic Formula with SAT-Style Questions
Use the quadratic formula when factoring is not simple.
Solve .
For , how many real solutions are there?
Solve .
If , which value of is a solution?
Key Takeaways to Remember for Quadratic Formula
- The quadratic formula always works when factoring fails.
- The discriminant tells you how many real solutions exist.
- Desmos provides quick decimal checks for your results.
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