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3.6 – The Quadratic Formula

Objectives

  • Write the general form of the quadratic formula and state its use.
  • State the conditions that must be met to use the quadratic formula.
  • Identify equations that can be solved using the quadratic formula.
  • Use the quadratic formula to find the solutions to a quadratic equation.
  • Define the discriminant and state what it tells about the solutions to the equation.
  • Apply the quadratic formula to find the roots of a quadratic equation.
  • Use the discriminant to determine how many roots a polynomial has.

 

Key Terms

  • Complex Numbers – The set of all numbers of the form a + bi, where a and b are any real numbers and i equals \sqrt{-1}.
  • Discriminant – The number under the radical sign in the quadratic formula. It is given by the expression b^2-4ac.
  • Imaginary Number – The square root of a negative number. The imaginary number i is defined as the square root of –1.
    • You can use i=\sqrt{-1} to find the square root of any negative number.
  • Quadratic Formula – A formula used to solve quadratic equations.
    • The solutions to the quadratic equation ax^2+bx+c=0 are given by the formula x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}.
  • Radical – A term that has the form \sqrt[n]{b} where b can be a number, a variable, or an expression that contains both.
    • The symbol \sqrt[n]{} indicates the nth root.
    • The symbol \sqrt{} (no n) indicates a square root.
  • Roots – Values for which a function equals zero.
    • The roots are also called zeros of the function.
    • Any x-value at which the graph of a function crosses the x-axis is a root of the function.

 

Notes

Deriving the Quadratic Formula
  • These are the steps to derive the quadratic formula.
    • If you can derive this without your notes on December 1st, you get an excused grade on a TST (meaning that your lowest TST grade is dropped)!
    • You will get ONE chance, and you cannot miss a step!

Alg2A 3.06 - Derive Quad Form

Solving Quadratic Equations
  • To solve quadratic equations, you can:
    • factor and use the zero product rule (Unit 3.4)
    • use algebra tiles (Unit 3.1)
    • use the guess-and-check method (esp when you have a coefficient in front of the x^2-term (Unit 3.2)
    • complete the square (Unit 3.5)
    • use the quadratic formula (This section: Unit 3.6)
  • The following conditions MUST be met before you can use the quadratic formula to find the solutions of an equation:
    • There can be no term whose degree is higher than 2.
    • The coefficient of the x^2-term can’t be zero.
    • One side of the equation must be zero.
  • To begin solving with the quadratic equation, simplify the equation and convert it into standard form: ax^2+bx+c=0.

Alg2A 3.06 - Standard FormAlg2A 3.06 - abc

Alg2A 3.06 - Can You Derive

  • Remember, the square root of a number has both, a positive and a negative answer.
    • Ex: \sqrt{16}=\pm4 because 4^2=16 and (-4)^2=16.
Discriminants and Complex Numbers
Alg2A 3.06 - Parts of Quadratic
Alg2A 3.06 - Imaginary
Roots
  • To Find the Roots of a Polynomial’s Graph:
    1. Start with the equation for a graphed polynomial in this form y = ax2 + bx + c.
    2. Set y equal to zero.
    3. Use the quadratic formula to solve for x.
    4. Write each solution as a point (x, 0).
    5. Those two points are the roots of the polynomial!

Alg2A 3.06 - Roots of a Polynomial

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