write down the quadratic formula for (100-theta)
Answers
Write a quadratic equation in standard form and identify the values of a, b, and c in a standard form quadratic equation.
· Use the Quadratic Formula to find all real solutions.
· Use the Quadratic Formula to find all complex solutions.
· Compute the discriminant and state the number and type of solutions.
· Solve application problems requiring the use of the Quadratic Formula.
Introduction
You can solve any quadratic equation by completing the square—rewriting part of the equation as a perfect square trinomial. If you complete the square on the generic equation ax2 + bx + c = 0 and then solve for x, you find that . This equation is known as the Quadratic Formula.
This formula is very helpful for solving quadratic equations that are difficult or impossible to factor, and using it can be faster than completing the square. The Quadratic Formula can be used to solve any quadratic equation of the form ax2 + bx + c = 0.
Standard Form
The form ax2 + bx + c = 0 is called standard form of a quadratic equation. Before solving a quadratic equation using the Quadratic Formula, it's vital that you be sure the equation is in this form. If you don't, you might use the wrong values for a, b, or c, and then the formula will give incorrect solutions.
Example
Problem
Rewrite the equation 3x + 2x2 + 4 = 5 in standard form and identify a, b, and c.
3x + 2x2 + 4 = 5
3x + 2x2 + 4 – 5 = 5 – 5
First be sure that the right side of the equation is 0. In this case, all you need to do is subtract 5 from both sides.
3x + 2x2 – 1 = 0
2x2 + 3x – 1 = 0
Simplify, and write the terms with the exponent on the variable in descending order.
2x2
+
3x
–
1
=
0
↓
↓
↓
ax2
bx
c
a = 2, b = 3, c = −1
Now that the equation is in standard form, you can read the values of a, b, and c from the coefficients and constant. Note that since the constant 1 is subtracted, c must be negative.
Answer
2x2 + 3x – 1 = 0; a = 2, b = 3, c = −1
Example
Problem
Rewrite the equation 2(x + 3)2 – 5x = 6 in standard form and identify a, b, and c.
2(x + 3)2 – 5x = 6
2(x + 3)2 – 5x – 6 = 6 – 6
First be sure that the right side of the equation is 0.
2(x2 + 6x + 9) – 5x – 6 = 0
2x2 + 12x + 18 – 5x – 6 = 0
2x2 + 12x – 5x + 18 – 6 = 0
2x2 + 7x + 12 = 0
Expand the squared binomial, then simplify by combining like terms.
Be sure to write the terms with the exponent on the variable in descending order.
2x2
+
7x
+
12
=
0
↓
↓
↓
a
b
c
a = 2, b = 7, c = 12
Now that the equation is in standard form, you can read the values of a, b, and c from the coefficients and constant.
Answer
2x2 + 7x + 12 = 0; a = 2, b = 7, c = 12
Identify the values of a, b, and c in the standard form of the equation 3x + x2 = 6.
A) a = 3, b = 1, c = 6
B) a = 1, b = 3, c = 6
C) a = 1, b = 3, c = −6
D) a = 3, b = 1, c = −6
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Deriving the Quadratic Formula
Let's complete the square on the general equation and see exactly how that produces the Quadratic Formula. Recall the process of completing the square.
· Start with an equation of the form x2 + bx + c = 0.
· Rewrite the equation so that x2 + bx is isolated on one side.
· Complete the square by adding to both sides.
· Rewrite the perfect square trinomial as a square of a binomial.
· Use the Square Root Property and solve for x.
Can you complete the square on the general quadratic equation ax2 + bx + c = 0? Try it yourself before you continue to the example below. Hint: Notice that in the general equation, the coefficient of x2 is not equal to 1. You can divide the equation by a, which makes some of the expressions a bit messy, but if you are careful, everything will work out, and at the end, you’ll have the Quadratic Formula!
Example
Problem
Complete the square of ax2 + bx + c = 0 to arrive at the Quadratic Formula.
Divide both sides of the equation by a, so that the coefficient of x2 is 1.
Rewrite so the left side is in form x2 + bx (although in this case bx is actually ).
Since the coefficient on x is , the value to add to both sides is .
Write the left side as a binomial squared.
Evaluate as .
Write the fractions on the right side using a common denominator.
Add the fractions on the right.
Use the Square Root Property. Remember that you want both the positive and negative square roots!
Subtract from both sides to isolate x.
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The denominator under the radical is a perfect square, so:
Add the fractions since they have a common denominator.