Math · Advanced Math

Quadratic equations — Digital SAT Math practice

Factoring, the quadratic formula, completing the square, roots.

Key terms

Quadratic Equation: Polynomial equation of degree two, typically written as ax^2 + bx + c = 0 where a ≠ 0.
Discriminant: The value b^2 - 4ac under √; determines if roots are real or complex.
Vertex: The highest or lowest point on a parabola, found by x = -b/(2a) and substituting back.
Standard Form: The arrangement ax^2 + bx + c = 0 where coefficients a, b, and c are real numbers.
Factoring: Rewriting a quadratic as a product of two binomials to apply the zero-product property.
Zeros: The x-values where the graph crosses the horizontal axis; identical to the roots of the equation.

Forgetting to set the equation equal to zero before factoring or using the quadratic formula.
Setting a factored product equal to a nonzero number and then each factor to that number; zero-product needs 0.
Using b instead of -b at the start of the quadratic formula.
Treating the discriminant as the roots; it only tells you how many real roots there are.

A student solves (x − 2)(x + 3) = 6 by writing x − 2 = 6 or x + 3 = 6. What went wrong?

The zero-product property only works when the product equals 0. Expand and subtract 6: x² + x − 12 = 0, so (x + 4)(x − 3) = 0 and x = −4 or x = 3.

Watch out: Setting each factor equal to the nonzero right side. Move everything to one side so the other side is exactly 0.

Use the quadratic formula to solve 3x² − 7x + 2 = 0. Show the substituted expression before simplifying.

x = (7 ± √((−7)² − 4(3)(2))) / (2(3)) = (7 ± √25) / 6, so x = 2 or x = 1/3.

Watch out: Dropping the negative sign on b, or dividing only part of the numerator by 2a.

Convert x² + 10x + 7 into vertex form by completing the square. What constant must be added and subtracted?

Add and subtract (10/2)² = 25, giving (x + 5)² + 7 - 25 or (x + 5)² - 18.

Watch out: The added constant is always (b/2)², not b² or 2b.

Without solving, find the sum and product of the roots for 4x² - 11x + 3 = 0. Which method saves time?

Use Vieta formulas: sum = -(-11)/4 = 11/4, product = 3/4. This avoids factoring or using the quadratic formula.

Watch out: The sum formula requires a negative sign for b; students often forget it and write b/a instead of -b/a.

If a quadratic function f(x) = x² - 9 has zeros at x = 3 and x = -3, what are the coordinates of its x-intercepts?

The x-intercepts are exactly (3, 0) and (-3, 0). Zeros give the x-coordinate, while intercepts include the y-coordinate of zero.

Watch out: Confusing the zero (a number) with the intercept (a point on the graph). Always pair the zero with a y-value of 0.

What is the sum of the solutions of x² − 9x + 18 = 0?

Show answer and explanation

Answer: 9

Factor: (x − 3)(x − 6) = 0, so x = 3 or x = 6. Sum = 9. (Shortcut: sum of roots = −b/a = 9.)

What is the greater solution of x^2 - 5x + 6 = 0?

Show answer and explanation

Answer: 3

x^2 - 5x + 6 = (x - 2)(x - 3) = 0, so x = 2 or x = 3; the greater is 3.

What is the vertex of the parabola y = x^2 + 6x + 5?

Show answer and explanation

Answer: (-3, -4)

y = (x + 3)^2 - 4, so the vertex is (-3, -4).