Quadratic equations are a key part of algebra, showing up in many real-world problems. The Square Root Property is a handy tool for solving these equations, especially when they're in certain forms.

This method helps us find the roots of quadratic equations, which tell us where a parabola crosses the x-axis. Understanding this property builds a strong foundation for more complex algebraic concepts.

Solving Quadratic Equations Using the Square Root Property

Square Root Property for ax² = k

• States if $a^2 = b$, then $a = \pm \sqrt{b}$ ($a$ can be positive or negative)
• To solve $ax^2 = k$:
  • Isolate $x^2$ term on one side of equation
  • Divide both sides by coefficient $a$ to get $x^2 = \frac{k}{a}$
  • Take square root of both sides: $x = \pm \sqrt{\frac{k}{a}}$
• Simplify square root if possible (perfect squares, simplify fractions)
• Solution will have two roots: one positive and one negative ($\pm$ symbol)
• Examples:
  • $4x^2 = 36 \rightarrow x^2 = 9 \rightarrow x = \pm 3$
  • $5x^2 = 45 \rightarrow x^2 = 9 \rightarrow x = \pm \sqrt{9} = \pm 3$
• The expression under the square root sign is called the radicand

Square Root Property with a(x - h)²

• Equations in form $a(x - h)^2 = k$ can be solved using Square Root Property
• First isolate $(x - h)^2$ term on one side of equation
• Divide both sides by coefficient $a$ to get $(x - h)^2 = \frac{k}{a}$
• Take square root of both sides: $x - h = \pm \sqrt{\frac{k}{a}}$
• Solve for $x$ by adding $h$ to both sides: $x = h \pm \sqrt{\frac{k}{a}}$
• Solution will have two roots: one where square root term is added to $h$ and one where it is subtracted from $h$
• Examples:
  • $3(x - 2)^2 = 75 \rightarrow (x - 2)^2 = 25 \rightarrow x - 2 = \pm 5 \rightarrow x = 2 \pm 5 = 7$ or $-3$
  • $2(x + 1)^2 = 18 \rightarrow (x + 1)^2 = 9 \rightarrow x + 1 = \pm 3 \rightarrow x = -1 \pm 3 = 2$ or $-4$
• The value of $h$ represents the x-coordinate of the axis of symmetry for the parabola

Interpreting quadratic equation solutions

• Real roots:
  • If discriminant ($b^2 - 4ac$) is positive, equation has two distinct real roots
  • If discriminant is zero, equation has one real root (double root)
• Irrational roots:
  • If discriminant is positive but not a perfect square, roots will be irrational (cannot be expressed as simple fraction or integer)
  • Can be left in simplest radical form or approximated as decimals ($\sqrt{2} \approx 1.414$)
• Complex roots:
  • If discriminant is negative, equation has two complex roots
  • In form $a + bi$, where $a$ is real part and $bi$ is imaginary part
  • $i$ is imaginary unit, defined as $i^2 = -1$ (square root of -1)
• Number of real roots provides insights into graph of quadratic function:
  • Two real roots: graph crosses x-axis at two points
  • One real root: graph touches x-axis at one point (vertex)
  • No real roots: graph never intersects x-axis

Additional Methods and Considerations

• The quadratic formula can be used to solve any quadratic equation in standard form (ax² + bx + c = 0)
• When solving equations, be cautious of extraneous solutions that may arise from algebraic manipulations
• The graph of a quadratic equation is always a parabola, with its vertex located on the axis of symmetry