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2 years ago

Solve and simplify: x(x - 1) = 121 – x

Mathematics

2 answers:

vaieri [72.5K]2 years ago

4 0

Answer:

$\boxed{\bold{x \ = \ 11 \ or \ x \ = \ -11}}$

Explanation:

$\bold{x(x \ - \ 1) \ = \ 121 \ - \ x}$

Simplify both sides of the equation

x² − x = −x + 121

Add x to both sides

x² − x + x = −x + 121 + x

x² = 121

Take square root

x = ± √121

x = 11 or x = −11

GuDViN [60]2 years ago

3 0

Answer:

$x=11$

Step-by-step explanation:

$x(x-1)=121-x\\\\x^2-x=121-x\\\\x^2=121\\\\x=\sqrt{121}\\\\x=11$

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The center of a circle is at the origin on a coordinate grid. The vertex of a parabola that opens upward is at (0, 9). If the ci

zhannawk [14.2K]

Answer:

"The maximum number of solutions is one."

Step-by-step explanation:

Hopefully the drawing helps visualize the problem.

The circle has a radius of 9 because the vertex is 9 units above the center of the circle.

The circle the parabola intersect only once and cannot intercept more than once.

The solution is "The maximum number of solutions is one."

Let's see if we can find an algebraic way:

The equation for the circle given as we know from the problem without further analysis is so far $x^2+y^2=r^2$ .

The equation for the parabola without further analysis is $y=ax^2+9$ .

We are going to plug $ax^2+9$ into $x^2+y^2=r^2$ for $y$ .

$x^2+y^2=r^2$

$x^2+(ax^2+9)^2=r^2$

To expand $(ax^2+9)^2$ , I'm going to use the following formula:

$(u+v)^2=u^2+2uv+v^2$ .

$(ax^2+9)^2=a^2x^4+18ax^2+81$ .

$x^2+y^2=r^2$

$x^2+(ax^2+9)^2=r^2$

$x^2+a^2x^4+18ax^2+81=r^2$

So this is a quadratic in terms of $x^2$

Let's put everything to one side.

Subtract $r^2$ on both sides.

$x^2+a^2x^4+18ax^2+81-r^2=0$

Reorder in standard form in terms of x:

$a^2x^4+(18a+1)x^2+(81-r^2)=0$

The discriminant of the left hand side will tell us how many solutions we will have to the equation in terms of $x^2$ .

The discriminant is $B^2-4AC$ .

If you compare our equation to $Au^2+Bu+C$ , you should determine $A=a^2$

$B=(18a+1)$

$C=(81-r^2)$

The discriminant is

$B^2-4AC$

$(18a+1)^2-4(a^2)(81-r^2)$

Multiply the (18a+1)^2 out using the formula I mentioned earlier which was:

$(u+v)^2=u^2+2uv+v^2$

$(324a^2+36a+1)-4a^2(81-r^2)$

Distribute the 4a^2 to the terms in the ( ) next to it:

$324a^2+36a+1-324a^2+4a^2r^2$

$36a+1+4a^2r^2$

We know that $a>0$ because the parabola is open up.

We know that $r>0$ because in order it to be a circle a radius has to exist.

So our discriminat is positive which means we have two solutions for $x^2$ .

But how many do we have for just $x$ .

We have to go further to see.

So the quadratic formula is:

$\frac{-B \pm \sqrt{B^2-4AC}}{2A}$

We already have $B^2-4AC}$

$\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}$

This is t he solution for $x^2$ .

To find $x$ we must square root both sides.

$x=\pm \sqrt{\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}}$

So there is only that one real solution (it actually includes 2 because of the plus or minus outside) here for x since the other one is square root of a negative number.

That is,

$x=\pm \sqrt{\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}}$

means you have:

$x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}}$

$x=\pm \sqrt{\frac{-(18a+1)-\sqrt{36a+1+4a^2r^2}}{2a^2}}$ .

The second one is definitely includes a negative result in the square root.

18a+1 is positive since a is positive so -(18a+1) is negative

2a^2 is positive (a is not 0).

So you have (negative number-positive number)/positive which is a negative since the top is negative and you are dividing by a positive.

We have confirmed are max of one solution algebraically. (It is definitely not 3 solutions.)

If r=9, then there is one solution.

If r>9, then there is two solutions as this shows:

$x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}}$

r=9 since our circle intersects the parabola at (0,9).

Also if (0,9) is intersection, then

$0^2+9^2=r^2$ which implies r=9.

Plugging in 9 for r we get:

$x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2(9)^2}}{2a^2}}$

$x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+324a^2}}{2a^2}}$

$x=\pm \sqrt{\frac{-(18a+1)+\sqrt{(18a+1)^2}}{2a^2}}$

$x=\pm \sqrt{\frac{-(18a+1)+18a+1}{2a^2}}$

$x=\pm \sqrt{\frac{0}{2a^2}}$

$x=\pm 0$

$x=0$

The equations intersect at x=0. Plugging into $y=ax^2+9$ we do get $y=a(0)^2+9=9$ .

After this confirmation it would be interesting to see what happens with assume algebraically the solution should be (0,9).

This means we should have got x=0.

$0=\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}$

A fraction is only 0 when it's top is 0.

$0=-(18a+1)+\sqrt{36a+1+4a^2r^2}$

Add 18a+1 on both sides:

$18a+1=\sqrt{36a+1+4a^2r^2$

Square both sides:

$324a^2+36a+1=36a+1+4a^2r^2$

Subtract 36a and 1 on both sides:

$324a^2=4a^2r^2$

Divide both sides by $4a^2$ :

$81=r^2$

Square root both sides:

$9=r$

The radius is 9 as we stated earlier.

Let's go through the radius choices.

If the radius of the circle with center (0,0) is less than 9 then the circle wouldn't intersect the parabola. So It definitely couldn't be the last two choices.

7 0

2 years ago

Read 2 more answers

Triangle X Y Z is shown. Angle X Y Z is 51 degrees and angle Y Z X is 76 degrees. The length of X Z is 2.6, the length of X Y is

stepan [7]

Answer:

$z=3.2\ units$

Step-by-step explanation:

we know that

In the triangle XYZ

Applying the law of sines

$\frac{2.6}{sin(51^o)}=\frac{z}{sin(76^o)}$

solve for z

$z=\frac{2.6}{sin(51^o)}sin(76^o)$

$z=3.2\ units$

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2 years ago

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If 20% of all manually filed returns contain errors, and 0.05% of all electronically filed returns contain errors, how much more

Marizza181 [45]

To get the correct answer to this question, you just need to divide 20% by 0.05% in order to get the answer, how many times a manual filer is more likely to make an error than an electronic file.
20: 0.05 = 400
The answer is C. 400 times more likely.

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2 years ago

Which of the following statements about 42xy – 49x + 30y – 35 are true? Check all of the boxes that apply. One of the factors is

kipiarov [429]

Answer:

One of the factors is (7x + 5).

One of the factors is (6y – 7).

Step-by-step explanation:

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2 years ago

A house on the land measures 45 feet by 38 feet,a wooded area covers 118 feet by 60 feet,and the front yard is 78 feet by 40 fee

den301095 [7]

Rectanglearea=legnth timees width
triagnlearea=1/2base times height

ok so we find volume of entier land first

we can cut into triangle an rectangle

triangle=height=650-200=450
base=600
area=1/2*600*450=135000 ft^2

rectangle=200 times 600=120000

add
135000+120000=255000ft^2

find other areas and subtract
areahouse=45*38=1710
areawooded=118*60=7080
frontyard=78*40=3120
add
1710+7080+3120=11910

minus

255000-11910=243090

how many acres are farmed

43560ft^2=1acre
divide both sides by 43560
1ft^2=1/43560 acre

multiply both sides by 243090
243090ft^2=243090/43560acre=5.5805acre=5.6acre rounded

answer is 5.6 acer

6 0

2 years ago

Solve and simplify: x(x - 1) = 121 – x​

Solve and simplify: x(x - 1) = 121 – x