2.5 Quadratic Equations

Quadratic equations are a key part of algebra, describing relationships where one variable is squared. They pop up in many real-world situations, from physics to economics. Understanding how to solve them is crucial for tackling more complex math problems.

There are several ways to crack these equations, including factoring, using the square root property, and applying the quadratic formula. Each method has its strengths, and knowing when to use which can save you time and headaches in problem-solving.

Solving Quadratic Equations

Factoring techniques for quadratics

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• Quadratic equations in standard form $ax^2 + bx + c = 0$ where $a$ is not equal to 0 ($a \neq 0$)
• Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero ($ab = 0$, then either $a = 0$ or $b = 0$, or both)
• Factoring by grouping involves combining like terms and factoring out common factors
  • Group terms with a common factor and factor out the greatest common factor (GCF) from each group
  • Factor out the GCF from the entire expression if possible ($6x^2 + 3x$ can be factored as $3x(2x + 1)$)
• Special factoring patterns include difference of squares and perfect square trinomials
  • Difference of squares formula $a^2 - b^2 = (a+b)(a-b)$ can be used to factor expressions like $x^2 - 9 = (x+3)(x-3)$
  • Perfect square trinomials $a^2 + 2ab + b^2 = (a+b)^2$ and $a^2 - 2ab + b^2 = (a-b)^2$ factor into the square of a binomial ($x^2 + 6x + 9 = (x+3)^2$)
• Solving quadratic equations by factoring involves factoring the expression and setting each factor equal to zero
  • Factor the quadratic expression completely
  • Set each factor equal to zero and solve the resulting linear equations to find the solutions ($x^2 - 5x + 6 = 0$ factors to $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$)
  • These solutions are also known as the roots of the quadratic equation

Square root property in equations

• Square root property states that if $x^2 = a$, then $x = \pm \sqrt{a}$
• Isolating the squared term involves adding or subtracting terms to get the squared term alone on one side of the equation ($x^2 + 4 = 20$ becomes $x^2 = 16$)
• Taking the square root of both sides of the equation and simplifying the result if possible ($\sqrt{x^2} = \sqrt{16}$ becomes $x = \pm 4$)
• Considering both positive and negative solutions is necessary because a squared term can have two square roots ($\sqrt{9} = \pm 3$)

Completing the square method

• Completing the square involves rewriting the quadratic equation in the form $x^2 + bx = -c$
  • Divide the coefficient of $x$ by 2 and square the result $(\frac{b}{2})^2$
  • Add and subtract $(\frac{b}{2})^2$ to the equation to create a perfect square trinomial
  • Factor the perfect square trinomial and isolate the squared term
  • Apply the square root property to solve for $x$ ($x^2 + 6x + 5 = 0$ becomes $(x+3)^2 = 4$, so $x = -3 \pm \sqrt{4} = -3 \pm 2$)
• Vertex form of a quadratic equation $y = a(x-h)^2 + k$ where $(h, k)$ is the vertex and $x = h$ is the axis of symmetry
• Graphing quadratic equations involves identifying the vertex and axis of symmetry, plotting additional points, and connecting them to form a parabola
  • Use the equation or a table of values to find points on the graph
  • The parabola will be symmetric about the axis of symmetry and open upward if $a > 0$ or downward if $a < 0$

Quadratic Formula and Applications

Quadratic formula applications

• Quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ can be used to find solutions for any quadratic equation $ax^2 + bx + c = 0$
• Discriminant $\Delta = b^2 - 4ac$ determines the nature of the solutions
  • If $\Delta > 0$, the equation has two distinct real solutions ($x^2 - 5x + 6 = 0$ has $\Delta = 1$, so $x = 2$ or $x = 3$)
  • If $\Delta = 0$, the equation has one repeated real solution ($x^2 - 6x + 9 = 0$ has $\Delta = 0$, so $x = 3$)
  • If $\Delta < 0$, the equation has no real solutions, only complex solutions ($x^2 + 2x + 5 = 0$ has $\Delta = -16$, so no real solutions)
• Simplifying the solutions by reducing square roots and fractions if possible
• Applications of quadratic equations involve solving word problems related to projectile motion, area, and other quadratic relationships
  • Identify the appropriate equation to model the situation, such as the height of a thrown ball $h(t) = -16t^2 + 64t + 5$
  • Interpret the solutions in the context of the problem, such as the time at which the ball reaches its maximum height or hits the ground

Additional Concepts in Quadratic Equations

• A quadratic equation is a specific type of polynomial function with a highest exponent of 2
• The coefficients in a quadratic equation are the numerical values that multiply the variables
• Graphing quadratic equations results in a parabola, which is a U-shaped curve
• The solutions to a quadratic equation can be found by solving the equation or by identifying where the graph of the function crosses the x-axis