Mathematics

Paper 1

# Quadratics

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Quadratics

**Quadratic Equations and Functions**

- A
**quadratic equation**takes the form*ax^2 + bx + c = 0*, where*a ≠ 0*. - The
**solutions**to a quadratic equation are given by the quadratic formula:*x = [-b ± sqrt(b^2 - 4ac)] / (2a)*. - The "
**discriminant**" is the term*b^2 - 4ac*. Its value determines the number of roots the equation has. - The
**graph**of a quadratic function is called a parabola. It is symmetric about the line*x = -b/2a*, known as the axis of symmetry. - The lowest or highest point on the graph of a quadratic function is called the
**vertex**. For a function given in the form*y = ax^2 + bx + c*, the vertex occurs at*x = -b/2a*. - The
**y-intercept**of a quadratic function is the value of*c*(since*y = c*when*x = 0*). - The
**x-intercepts**or roots of a quadratic function are the solutions to the equation*ax^2 + bx + c = 0*.

**Factoring Quadratics**

- Factoring is a method used to solve a quadratic equation. An equation can be factored if it can be rewritten in the form
*(dx + e)(fx + g) = 0*, where*d*and*f*are the factors of*a*, and*e*and*g*are the factors of*c*that add up to*b*. - The
**zero-product property**states that if a product of factors equals zero, then at least one of the factors must be zero. This property allows us to set each factor equal to zero and solve for*x*. - Factoring is not always possible, particularly in cases where the discriminant is negative, expressing that the equation has no real roots.

**Quadratic Inequalities**

- Quadratic inequalities take the form
*ax^2 + bx + c < 0*,*ax^2 + bx + c ≤ 0*,*ax^2 + bx + c > 0*, or*ax^2 + bx + c ≥ 0*. - Solve a quadratic inequality by first factoring the quadratic function and then determining the intervals where the function is positive or negative.
- Plot the graph; where the graph lies above the x-axis,
*y > 0*, and where the graph lies below the x-axis,*y < 0*.

**Completing the Square**

- Completing the square is another method used to solve a quadratic equation, especially useful when the equation does not factor easily.
- This process involves transforming the given equation into the form
*(x - h)^2 = k*, which allows us to identify the vertex '*h*' and y-intercept '*k*'. - After completing the square and simplifying the equation, solutions can be found by solving
*x - h = sqrt(k)*and*x - h = -sqrt(k)*.

**Quadratic Formula Derivation**

- The quadratic formula can be derived from the process of completing the square
- Start with the standard form of a quadratic equation (
*ax^2 + bx + c = 0*), and manipulate it to derive the formula: *x = [-b ± sqrt(b^2 - 4ac)] / 2a.