Quadratic Equations: Factoring and More

• Quadratic factorization, completing the square, graphing quadratics, and solving word problems are key concepts in understanding quadratic functions

Quadratic Factorization

• Quadratic factorization expresses a quadratic expression as a product of two linear factors.
• Factoring simplifies solving quadratic equations.
• The general form of a quadratic expression ( ax^2 + bx + c ), where ( a ), ( b ), and ( c ) are constants.

Simple Quadratic Expression:

• ( x^2 + 5x + 6 ) is an example.
• Find two numbers that multiply to 6 and add to 5.
• The numbers 2 and 3 satisfy these conditions.
• Therefore, ( x^2 + 5x + 6 = (x + 2)(x + 3) ).

Quadratic Expression with a Leading Coefficient:

• ( 2x^2 + 7x + 3 ) serves as an example.
• Multiply the leading coefficient (2) by the constant term (3) to get 6.
• Find two numbers that multiply to 6 and add to 7.
• The numbers 1 and 6 fit.
• Rewrite the middle term: ( 2x^2 + x + 6x + 3 )
• Factor by grouping: ( x(2x + 1) + 3(2x + 1) )
• Therefore, ( 2x^2 + 7x + 3 = (x + 3)(2x + 1) )

Completing the Square

• Completing the square converts a quadratic equation into a perfect square trinomial
• This simplifies solving the equation.
• A perfect square trinomial factors into the square of a binomial.
• For example, ( x^2 + 2ax + a^2 = (x + a)^2 )

Solving by Completing the Square:

• Take the equation ( x^2 + 6x + 5 = 0 ) as an example.
• Move the constant term: ( x^2 + 6x = -5 )
• Add ( (6/2)^2 = 9 ) to both sides: ( x^2 + 6x + 9 = -5 + 9 )
• Factor and simplify: ( (x + 3)^2 = 4 )
• Take the square root: ( x + 3 = \pm 2 )
• Solve for ( x ): ( x = -3 \pm 2 ), so ( x = -1 ) or ( x = -5 )

Leading Coefficient Not Equal to 1:

• When ( a \neq 1 ), divide the equation by ( a ) first.
• For example, for ( 2x^2 + 8x + 6 = 0 ), divide by 2 to get ( x^2 + 4x + 3 = 0 ), then complete the square.

Quadratic Graph

• The graph of ( f(x) = ax^2 + bx + c ) is a parabola.
• Key features of a parabola are its vertex, axis of symmetry, and intercepts.

Vertex:

• The vertex is the turning point of the parabola.
• It represents either the minimum or maximum point.
• The x-coordinate of the vertex is ( x = -\frac{b}{2a} )
• Substitute this ( x ) value into the function to find the ( y )-coordinate.

Axis of Symmetry:

• This vertical line passes through the vertex, dividing the parabola symmetrically.
• Its equation is ( x = -\frac{b}{2a} ).

Intercepts:

• The ( y )-intercept occurs where the parabola intersects the ( y )-axis.
• Setting ( x = 0 ) gives the ( y )-intercept: ( f(0) = c ).
• The ( x )-intercepts occur where the parabola intersects the ( x )-axis.
• Setting ( f(x) = 0 ) allows solving for ( x ) by factoring, completing the square, or using the quadratic formula.

Parabola Direction:

• If ( a > 0 ), the parabola opens upwards, and the vertex is a minimum.
• If ( a < 0 ), the parabola opens downwards, and the vertex is a maximum.

Solving Quadratic Word Problems

• Quadratic equations model projectile motion, area calculations, and optimization problems.

General Steps:

• Read the problem to understand the question, knowns, and unknowns.
• Define variables to represent unknown quantities.
• Set up a quadratic equation based on the problem.
• Solve the equation using factoring, completing the square, or the quadratic formula.
• Check that the solution is reasonable, discarding extraneous values.
• State the final answer with appropriate units.

Projectile Motion Example:

• A ball is thrown upwards at 20 m/s from 1 meter, with height ( h(t) = -5t^2 + 20t + 1 ).
• To find the maximum height, determine the vertex of the parabola.
• The ( t )-coordinate of the vertex is ( t = -\frac{20}{2(-5)} = 2 ) seconds.
• The maximum height is ( h(2) = -5(2)^2 + 20(2) + 1 = 21 ) meters.

Area Calculation Example:

• A garden's length is 5 meters more than its width; the area is 84 square meters.
• If ( w ) is the width, the length is ( w + 5 ).
• The area equation is ( w(w + 5) = 84 ).
• Expand to ( w^2 + 5w - 84 = 0 ).
• Factor to ( (w - 7)(w + 12) = 0 ).
• Since width cannot be negative, ( w = 7 ) meters, and the length is 12 meters.