Solving Quadratics with Square Roots

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 roots of a quadratic equation are the values of x that satisfy the equation, meaning they make the equation equal to zero.

To solve a quadratic equation using square roots, isolate the $$x^2$$ term and then take the square root of both sides, remembering to consider both the positive and negative roots.

The square root property states that if $$x^2 = k$$ , then $$x = \text{±}\sqrt{k}$$ .

$$x = \text{±} 1$$ .

The discriminant, given by $$D = b^2 - 4ac$$ , determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

$$x = \text{±} \sqrt{10}$$ .

It means that the graph of the quadratic function does not intersect the x-axis, which occurs when the discriminant is negative.

$$x = \text{±} 8$$ .

The vertex is the highest or lowest point of the parabola represented by the quadratic function, found at the point $$(h, k)$$ where $$h = -\frac{b}{2a}$$ and $$k = f(h)$$ .