Download 1. What is the standard form of a quadratic equation?

1. What is the standard form of a quadratic equation?
ax 2 + bx + c = 0
Is the following quadratic equation:
-3x² + 30x - 15 = 0
in standard form? Justify your answer. If it is not in standard form, what will be an equivalent quadratic equation in
standard form? Is there only one quadratic equation in standard form equivalent to the equation above? If so,
justify. If not, provide examples of at least two quadratic equations in standard form equivalent to the above
equation.
Yes it is in standard form. a = −3, b = 30, and c = −15 . There are other equations in standard form that are
equivalent to this. These can be found by multiplying the equation (both sides) by a constant factor. Two examples
of equivalent equations in standard form are
3x 2 − 30x + 15 = 0
x 2 − 10x + 5 = 0
What is factoring and the principle of zero products, and how can these concepts be applied to solving quadratic
equations?
Factoring a polynomial means finding a set of lower order polynomials whose product gives the original
polynomial. The principle of zero products means that the product of terms to equal zero if and only if one or more
of the terms is zero. In solving quadratic equations, if we factor the quadratic into two linear terms, then the
solutions to the quadratic are the solutions found by setting each of the linear terms equal to zero.
What is the rationale behind "completing the square" method? Can this method be used with any quadratic
equation? If not, provide an example of an equation that can't be solved using "completing the square".
Completing the square means adding a constant to each side of a standard quadratic equation so that the resulting
quadratic expression is the square of a linear binomial. Taking the square root of both sides then yields the
solutions to the quadratic equation. This method can be used with any quadratic equation.
Come up with a quadratic equation for your classmates to solve using the "completing the square" method - make
sure that you write it in standard form and that the coefficients a, b, c are non-zero and all different.
solve
x 2 − 14x + 47 = 0
solution
x 2 − 14x + 47 = 0
x 2 − 14x + 49 = 2
(x − 7)2 = 2
x−7= ± 2
{
x = 7 − 2, 7 + 2
}
2. SOLVE.
x=
20 ± 202 − 4(5)(35)
= 2 ± 3i
2(5)
3. SOLVE.
x=
−18 ± 182 − 4(−1)(−19)
= 9 ± 62
−2
4. SOLVE.
−12 ± 122 − 4(30)
x=
= −6 ± 6
2
5. SOLVE.
5x^2+12x+12=0
−12 ± 122 − 4(5)(12) −6 ± 2 −6
x=
=
2(5)
5
6. SOLVE.
(x − 2)(x − 4) = 0
x = 2, 4
7. SOLVE.
x=
−24 ± 242 − 4(4)(14) −6 ± 22
=
2(4)
2
8. What is the "Quadratic Formula"? How does it help us solve quadratic equations? How does the quadratic formula
relate to the x-intercepts of the graph of the appropriate quadratic function? If a graph of the quadratic function lies
fully above or fully below the x-axis, what can we say about the expression: b²-4ac (where a, b, and c are standard
coefficients of this function)?
−b ± b2 − 4ac
The quadratic formula is
. This gives us the solutions to any quadratic, ax 2 + bx + c = 0 . The
2a
solutions of this are the x-intercepts of the graph of the function y = ax 2 + bx + c . If the graph lies fully above or
below the x-axis, then b2 − 4ac is negative.
What is the discriminant? Why does the textbook recommend to compute the discriminant first when solving the
quadratic equations? Does the discriminant tell us anything about the number of real solution to a given quadratic
equation, and if so, what does it tell us?
The discriminant is b2 − 4ac . If this is negative, we know thre are two imaginary solutions and no real solutions.
If it is positive, there are two real solutions. If it’s zero, there is only one solution, which is real.
What are "equations quadratic in form"? Do you expect equations quadratic in form to have more, less, or the same
number of solutions compared to a quadratic equation? Provide an example of an equation quadratic in form
for your classmates to solve, making sure it has at least one real solution.
An equation in quadratic form is the same as a quadratic equation, but the variable may itself be a more complex
expression. Thus the equation can be of higher order than a quadratic, and have correspondingly more solutions.
For example
3( y 2 )2 − 7( y 2 ) + 4 = 0
( y 2 − 1)(3y 2 − 4) = 0
y 2 = 1,
4
3
y = ±1, ±
2
3
9. How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic
equation if you are only given the solution? Is it possible to have different quadratic equations with the same
solution? Explain. Provide your classmate’s with one or two solutions with which they must create a quadratic
equation.
If the discriminant, b2 − 4ac , is negative, we know thre are two imaginary solutions and no real solutions. If it is
positive, there are two real solutions. If it’s zero, there is only one solution, which is real.
The solutions to the quadratic equation are the zeros of the graph of that function. Two quadratic functions can
share one or even both intercepts, but not be identical. Example. Find a quadratic equation with solution
x = −3, 2
solution:
(x + 3)(x − 2) = 0
x2 + x − 6 = 0
10. Explain in your own words how the principle of square roots is used to solve quadratic equations. What form must
a quadratic equation be in to use the principle of square roots for solving? Demonstrate the process with your own
example.
The principle of square roots can be used if the quadratic equation can be manipulated into a form where it is a
perfect square. Then, taking the square root of both sides will directly give the solutions. Example
solve
4x 2 − 4x − 5 = 0
solution
4x 2 − 4x − 5 = 0
4x 2 − 4x + 1 = 6
(2x − 1)2 = 6
2x − 1 = ± 6
⎧⎪1 − 6 1 + 6 ⎫⎪
x=⎨
,
⎬
2
2 ⎪⎭
⎩⎪