A quadratic equation is a second-degree polynomial in the form ax2 + bx + c = 0, where a, b, and c are constants, and x is the variable.
Examples of quadratic equations:
The vertex form of a quadratic equation is written as:
y = a(x - h)2 + k, where (h, k) is the vertex of the parabola.
This form is useful for finding the vertex of the parabola directly and for graphing the equation.
The discriminant of a quadratic equation is the expression under the square root in the quadratic formula: b² - 4ac.
When the discriminant b² - 4ac is negative, the quadratic equation has no real solutions, but instead has two complex roots.
To find the complex roots, use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a, where the square root of a negative number gives an imaginary number, denoted as i.
Example: Solve x2 + 2x + 5 = 0:
Here, a = 1, b = 2, and c = 5. The discriminant is 2² - 4(1)(5) = 4 - 20 = -16. The roots are:
x = (-2 ± √-16) / 2, which simplifies to x = (-2 ± 4i) / 2, or x = -1 ± 2i.
Enter the coefficients of the quadratic equation ax² + bx + c:
This method involves expressing the quadratic expression as the product of two binomials and setting each equal to zero to find the solutions.
Steps:
ax² + bx + c = 0.c and add to b.Example: Solve x2 - 5x + 6 = 0 by factoring.
Solution: (x - 2)(x - 3) = 0, so x = 2 or x = 3.
This method involves creating a perfect square trinomial on one side of the equation, which can then be solved by taking the square root of both sides.
Steps:
ax² + bx = -c.a ≠ 1, divide each term by a to make the coefficient of x² equal to 1.(b/2)² to both sides to complete the square.(x + b/2)².x by taking the square root of both sides.Example: Solve x² - 6x + 5 = 0 by completing the square.
Solution: (x - 3)² = 4, so x = 3 ± 2, giving x = 5 or x = 1.
The quadratic formula is a general method applicable to any quadratic equation. It works by substituting the values of a, b, and c from the equation ax² + bx + c = 0 into the formula.
Formula: x = (-b ± √(b² - 4ac)) / 2a
Steps:
a, b, and c from the equation.x.Example: Solve 2x² - 4x - 6 = 0 using the quadratic formula.
Solution: x = (4 ± √(16 + 48)) / 4, so x = 3 or x = -1.
Quadratic equations are essential in physics, especially in analyzing projectile motion. The height of an object in free fall is often modeled by a quadratic equation:
h = vt - 0.5gt² where h is height, v is initial velocity, and g is the acceleration due to gravity.
Example: To find how long it takes for a ball to reach the ground when thrown upward, set h = 0 and solve the quadratic equation.
Companies use quadratic equations to determine the optimal number of products to produce to maximize profit. The profit function P(x) = ax² + bx + c is a parabola, where the maximum point gives the optimal production level.
Example: For a product with a cost function C(x) = 5x² + 50x + 200 and revenue function R(x) = 60x, the profit function is P(x) = R(x) - C(x).
Solving the equation for P'(x) = 0 (where P(x) is maximized) provides the optimal production level.
Quadratic equations are frequently used in architecture and engineering to calculate stresses on structures and determine ideal shapes and supports for load distribution.
Example: An engineer might use a quadratic function to determine the optimal arch shape for a bridge to distribute weight evenly.
What is the solution to x2 - 4x - 5 = 0?
1. Which method can be used to solve x2 - 5x + 6 = 0?
2. When using the method of completing the square for x2 - 4x + 4 = 0, what value should you add to complete the square?
3. What is the quadratic formula used to solve any quadratic equation?
Enter the coefficients of the quadratic equation ax² + bx + c:
Compound Angle Identities involve the addition or subtraction of two angles:
Double Angle Identities simplify calculations when dealing with twice an angle:
Visualize these identities by entering a range of angles: