Solving Quadratic Equations Using the Square Root Property

The square root property states that if $$x^2 = a$$ , then $$x = ±\text{sqrt}(a)$$ . This is used to find the solutions of quadratic equations.

To isolate a squared term, you can add or subtract other terms from both sides of the equation. For example, in $$(x-1)^2 = 32$$ , add 1 to both sides to isolate the squared term.

Multiply both sides by 2 to eliminate the fraction: $$(x-1)^2 = 64$$ .

$$x = ±4$$ .

It means that the equation does not intersect the x-axis, and the solutions are complex or imaginary numbers.

First, simplify to get $$-7x^2 = 63$$ , then $$x^2 = -9$$ , which has no real solutions.

$$x = 4$$ and $$x = -3$$ .

Take the square root of both sides, giving $$x = ±\text{sqrt}(a)$$ .

First, isolate the squared term to find $$x = -5$$ and $$x = -3$$ .

The discriminant, given by $$b^2 - 4ac$$ , determines the nature of the roots: if positive, two real solutions; if zero, one real solution; if negative, no real solutions.