CA/HW Day 5 Review Quadratic Equations and Zero-Product Property

A quadratic equation is a polynomial equation of degree 2, typically in the form $$ax^2 + bx + c = 0$$ , where a, b, and c are constants and a ≠ 0.

The Zero-Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. For example, if $$ab = 0$$ , then either $$a = 0$$ or $$b = 0$$ .

To solve a quadratic equation by factoring, express it in the form $$(x - p)(x - q) = 0$$ , then set each factor to zero: $$x - p = 0$$ and $$x - q = 0$$ .

The standard form of a quadratic equation is $$y = ax^2 + bx + c$$ , where a, b, and c are constants.

The roots of a quadratic equation are the values of x that make the equation equal to zero. They can be found using factoring, completing the square, or the quadratic formula.

The quadratic formula is used to find the roots of a quadratic equation and is given by $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$ .

To complete the square, rewrite the equation as $$(x - 3)^2 - 4 = 0$$ , then solve for x.

The vertex form of a quadratic equation is $$y = a(x - h)^2 + k$$ , where (h, k) is the vertex of the parabola.

The vertex of a parabola given in standard form $$y = ax^2 + bx + c$$ can be found using the formula $$h = -\frac{b}{2a}$$ and then substituting h back into the equation to find k.

A quadratic equation has no real solutions when the discriminant (b^2 - 4ac) is less than zero, indicating that the parabola does not intersect the x-axis.