Quadratic equations appear whenever a relationship curves instead of forming a straight line, such as projectile motion, area problems, and optimization. A quadratic equation has the standard form ax^2 + bx + c = 0, where a is not 0. The quadratic formula gives a reliable way to solve any quadratic equation, even when factoring is difficult or impossible.
The discriminant, b^2 - 4ac, quickly tells you what kind of solutions to expect before you finish solving.
Key Facts
- Standard form of a quadratic equation: ax^2 + bx + c = 0, with a != 0.
- Quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a.
- Discriminant: D = b^2 - 4ac.
- If D > 0, the quadratic has two distinct real roots.
- If D = 0, the quadratic has one repeated real root at x = -b / 2a.
- If D < 0, the quadratic has two complex roots and its graph does not cross the x-axis.
Vocabulary
- Quadratic equation
- An equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and a is not 0.
- Quadratic formula
- A formula that gives the solutions of any quadratic equation using the coefficients a, b, and c.
- Discriminant
- The expression b^2 - 4ac inside the square root of the quadratic formula that determines the number and type of roots.
- Root
- A value of x that makes the quadratic equation equal to 0.
- Parabola
- The U-shaped graph of a quadratic function y = ax^2 + bx + c.
Common Mistakes to Avoid
- Forgetting that a must not be 0, which is wrong because the equation would no longer be quadratic and the quadratic formula would not apply.
- Using b instead of -b at the start of the formula, which changes the signs of the answers and gives incorrect roots.
- Dropping the ± symbol, which is wrong because most quadratics with a positive discriminant have two different solutions.
- Calculating the discriminant as b^2 + 4ac, which is wrong because the correct expression is b^2 - 4ac and the sign determines the root type.
Practice Questions
- 1 Solve 2x^2 - 5x - 3 = 0 using the quadratic formula.
- 2 For x^2 + 6x + 10 = 0, calculate the discriminant and state whether the roots are real or complex.
- 3 A parabola opens upward and touches the x-axis at exactly one point. What does this tell you about the discriminant and the roots?