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ss7ja [257]
3 years ago
15

Complete the steps of derivation of the quadratic formula

Mathematics
1 answer:
Stells [14]3 years ago
3 0
Step 1. We are going to begin with standard form a quadratic equation: ax^2+bx+c=0

Step 2.  Divide both sides of the equation by a:
\frac{ax^2+bx+c}{a} = \frac{0}{a}
\frac{ax^2}{a} + \frac{bx}{c} + \frac{c}{a} =0
x^{2} + \frac{b}{a} x+ \frac{c}{a} =0

Step 3. Subtract \frac{c}{a} from both sides of the equation:
x^{2} + \frac{b}{a} x+ \frac{c}{a}- \frac{c}{a}  =0 - \frac{c}{a}
x^{2} + \frac{b}{a} x=- \frac{c}{a}

Step 4. Complete the square of the left hand side by adding ( \frac{b}{2a} )^2 to both sides:
x^{2} + \frac{b}{a} x+( \frac{b}{2a} )^2=- \frac{c}{a} +( \frac{b}{2a} )^2
(x+ \frac{b}{2a} )^2=- \frac{c}{a} +( \frac{b}{2a} )^2

Step 5. Take square root to both sides of the equation:
\sqrt{(x+ \frac{b}{2a} )^2} =+/- \sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}
x+ \frac{b}{2a}=+/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}

Step 6. Subtract \frac{b}{2a} from both sides of the equation:
x+ \frac{b}{2a}- \frac{b}{2a} =- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}
x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}

Step 7. Simplify the radicand of the left hand side of the equation:
x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} + \frac{b^2}{(2a)^2}}
x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} + \frac{b^2}{4a^2}}
x=- \frac{b}{2a} +/-\sqrt{- \frac{4ac}{4a^2} + \frac{b^2}{4a^2}}
x=- \frac{b}{2a} +/-\sqrt{- \frac{4ac+b^2}{4a^2} }

Step 8. Take 4a^2 outside the radical:
x=- \frac{b}{2a} +/- \frac{ \sqrt{-4ac+b^2} }{ \sqrt{4a^2} }
x=- \frac{b}{2a} +/- \frac{ \sqrt{-4ac+b^2} }{2a} }

Step 9. Combine the two fractions and <span>rearrange</span> the terms in the radical: 
x= \frac{-b+/- \sqrt{b^2-4ac} }{2a}
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The pth percentile is a value such that approximately a. p percent of the observations are less than the value and (100 - p) per
kogti [31]

Answer:

p percent of the observations are less than the value and (100 - p) percent are more than this value.

Step-by-step explanation:

Given : The pth percentile is a value such that approximately

Solution :

Definition : A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations falls

So, The pth percentile means p percentage of observations in a group of observations falls bellow the value

So, (100-p) percentage of observations in a group of observations falls above the value

So, Option a is true

Hence p percent of the observations are less than the value and (100 - p) percent are more than this value.

5 0
3 years ago
| 2x + 2y = 10<br> y = 5+x
melisa1 [442]
2x + 2(5+x)=10
2x+10+2x=10
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4x=0
X=0
6 0
3 years ago
Find the product. Simplify.<br> (a + 1)(a –5)=
Morgarella [4.7K]

Answer:

(a + 1)(a - 5) \\   {a}^{2}  - 5a +  a- 5 \\  {a}^{2}  - 4 a- 5

4 0
3 years ago
The graph shows the function f(x)=3^x
tangare [24]
The answer is D.
You have to make the whole numbers equal to then solve for the exponents:
F(x) means y, therefore, y is 9.
9 = 3^x.
3^2 is the same as 9.
Then, 3^2 = 3^x.
Then solve for the exponents,
2 = x
Therefore your answer is 2.
8 0
3 years ago
Graph the image of this figure after a dilation with a scale factor of 2 centered at (2, 2).
Maru [420]

Answer:

(-6,6), (0,12), (4,8)

Step-by-step explanation:

To dilate an object, we need to multiply the x and y values by the given scale factor.

In this case the scale factor is 2 --> 2(x, y)

Before-> After dilation

2(-3,3) = (-6,6)

2(0,6) = (0,12)

2(2,4) = (4,8)

Please leave a 'thanks' if this helps!

6 0
3 years ago
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