Question Solve the equation $x^{2}-10x-19 = (x-p)^{2}+q$ for $x$ , where $p$ and $q$ are constants.

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic equation in the form $x^{2}-10x-19$ .
2. We need to express this quadratic equation in the form $(x-p)^{2}+q$ .
3. We will use the method of completing the square to rewrite the given quadratic equation.

STEP 2

The general form of a quadratic equation is $ax^{2} + bx + c$ . To complete the square, the equation must be in the form $a(x-h)^{2} + k$ , where $h$ and $k$ are constants. We will follow this method to rewrite the given equation.

STEP 3

First, we identify the coefficient of $x$ in the given equation, which is $-10$ . We will use this to find the value of $p$ in the expression $(x-p)^{2}$ .

STEP 4

To complete the square, we take half of the coefficient of $x$ and square it. This will give us the value to add and subtract to complete the square.
$p = \frac{-(-10)}{2} = \frac{10}{2} = 5$

STEP 5

Now we square the value of $p$ to find the number to add and subtract from the equation.
$p^{2} = 5^{2} = 25$

STEP 6

We add and subtract $p^{2}$ to the equation $x^{2}-10x-19$ . We add it inside the parentheses to complete the square and subtract it outside to keep the equation balanced.
$x^{2}-10x + 25 - 25 - 19 = (x-p)^{2}+q$

STEP 7

Now we rewrite the equation, grouping the terms to form a perfect square and combining the constants.
$(x^{2}-10x + 25) - 25 - 19 = (x-5)^{2} - 25 - 19$

STEP 8

Combine the constant terms $-25$ and $-19$ .
$q = -25 - 19 = -44$

STEP 9

Now we have the equation in the form of $(x-p)^{2}+q$ .
$(x-5)^{2} - 44 = (x-p)^{2}+q$

STEP 10

We can now identify the values of $p$ and $q$ .
$p = 5$ $q = -44$
The equation $x^{2}-10x-19$ can be expressed as $(x-5)^{2}-44$ .

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Question Solve the equation $x^{2}-10x-19 = (x-p)^{2}+q$ for $x$ , where $p$ and $q$ are constants.

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic equation in the form $x^{2}-10x-19$ .
2. We need to express this quadratic equation in the form $(x-p)^{2}+q$ .
3. We will use the method of completing the square to rewrite the given quadratic equation.

STEP 2

The general form of a quadratic equation is $ax^{2} + bx + c$ . To complete the square, the equation must be in the form $a(x-h)^{2} + k$ , where $h$ and $k$ are constants. We will follow this method to rewrite the given equation.

STEP 3

First, we identify the coefficient of $x$ in the given equation, which is $-10$ . We will use this to find the value of $p$ in the expression $(x-p)^{2}$ .

STEP 4

To complete the square, we take half of the coefficient of $x$ and square it. This will give us the value to add and subtract to complete the square.
$p = \frac{-(-10)}{2} = \frac{10}{2} = 5$

STEP 5

Now we square the value of $p$ to find the number to add and subtract from the equation.
$p^{2} = 5^{2} = 25$

STEP 6

We add and subtract $p^{2}$ to the equation $x^{2}-10x-19$ . We add it inside the parentheses to complete the square and subtract it outside to keep the equation balanced.
$x^{2}-10x + 25 - 25 - 19 = (x-p)^{2}+q$

STEP 7

Now we rewrite the equation, grouping the terms to form a perfect square and combining the constants.
$(x^{2}-10x + 25) - 25 - 19 = (x-5)^{2} - 25 - 19$

STEP 8

Combine the constant terms $-25$ and $-19$ .
$q = -25 - 19 = -44$

STEP 9

Now we have the equation in the form of $(x-p)^{2}+q$ .
$(x-5)^{2} - 44 = (x-p)^{2}+q$

STEP 10

We can now identify the values of $p$ and $q$ .
$p = 5$ $q = -44$
The equation $x^{2}-10x-19$ can be expressed as $(x-5)^{2}-44$ .

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Question Solve the equation x2−10x−19=(x−p)2+qx^{2}-10x-19 = (x-p)^{2}+qx2−10x−19=(x−p)2+q for xxx, where ppp and qqq are constants.

Studdy Solution

STEP 1

Assumptions1. We are given a quadratic equation in the form x2−10x−19x^{2}-10x-19x2−10x−19.2. We need to express this quadratic equation in the form (x−p)2+q(x-p)^{2}+q(x−p)2+q.3. We will use the method of completing the square to rewrite the given quadratic equation.

STEP 2

The general form of a quadratic equation is ax2+bx+cax^{2} + bx + cax2+bx+c. To complete the square, the equation must be in the form a(x−h)2+ka(x-h)^{2} + ka(x−h)2+k, where hhh and kkk are constants. We will follow this method to rewrite the given equation.

STEP 3

First, we identify the coefficient of xxx in the given equation, which is −10-10−10. We will use this to find the value of ppp in the expression (x−p)2(x-p)^{2}(x−p)2.

STEP 4

To complete the square, we take half of the coefficient of xxx and square it. This will give us the value to add and subtract to complete the square.p=−(−10)2=102=5p = \frac{-(-10)}{2} = \frac{10}{2} = 5p=2−(−10)​=210​=5

STEP 5

Now we square the value of ppp to find the number to add and subtract from the equation.p2=52=25p^{2} = 5^{2} = 25p2=52=25

STEP 6

We add and subtract p2p^{2}p2 to the equation x2−10x−19x^{2}-10x-19x2−10x−19. We add it inside the parentheses to complete the square and subtract it outside to keep the equation balanced.x2−10x+25−25−19=(x−p)2+qx^{2}-10x + 25 - 25 - 19 = (x-p)^{2}+qx2−10x+25−25−19=(x−p)2+q

STEP 7

Now we rewrite the equation, grouping the terms to form a perfect square and combining the constants.(x2−10x+25)−25−19=(x−5)2−25−19(x^{2}-10x + 25) - 25 - 19 = (x-5)^{2} - 25 - 19(x2−10x+25)−25−19=(x−5)2−25−19

STEP 8

Combine the constant terms −25-25−25 and −19-19−19.q=−25−19=−44q = -25 - 19 = -44q=−25−19=−44

STEP 9

Now we have the equation in the form of (x−p)2+q(x-p)^{2}+q(x−p)2+q.(x−5)2−44=(x−p)2+q(x-5)^{2} - 44 = (x-p)^{2}+q(x−5)2−44=(x−p)2+q

STEP 10

We can now identify the values of ppp and qqq.p=5p = 5p=5 q=−44q = -44q=−44The equation x2−10x−19x^{2}-10x-19x2−10x−19 can be expressed as (x−5)2−44(x-5)^{2}-44(x−5)2−44.

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Question Solve the equation $x^{2}-10x-19 = (x-p)^{2}+q$ for $x$ , where $p$ and $q$ are constants.

Assumptions
1. We are given a quadratic equation in the form $x^{2}-10x-19$ .
2. We need to express this quadratic equation in the form $(x-p)^{2}+q$ .
3. We will use the method of completing the square to rewrite the given quadratic equation.

The general form of a quadratic equation is $ax^{2} + bx + c$ . To complete the square, the equation must be in the form $a(x-h)^{2} + k$ , where $h$ and $k$ are constants. We will follow this method to rewrite the given equation.

First, we identify the coefficient of $x$ in the given equation, which is $-10$ . We will use this to find the value of $p$ in the expression $(x-p)^{2}$ .

To complete the square, we take half of the coefficient of $x$ and square it. This will give us the value to add and subtract to complete the square.
$p = \frac{-(-10)}{2} = \frac{10}{2} = 5$

Now we square the value of $p$ to find the number to add and subtract from the equation.
$p^{2} = 5^{2} = 25$

We add and subtract $p^{2}$ to the equation $x^{2}-10x-19$ . We add it inside the parentheses to complete the square and subtract it outside to keep the equation balanced.
$x^{2}-10x + 25 - 25 - 19 = (x-p)^{2}+q$

Now we rewrite the equation, grouping the terms to form a perfect square and combining the constants.
$(x^{2}-10x + 25) - 25 - 19 = (x-5)^{2} - 25 - 19$

Combine the constant terms $-25$ and $-19$ .
$q = -25 - 19 = -44$

Now we have the equation in the form of $(x-p)^{2}+q$ .
$(x-5)^{2} - 44 = (x-p)^{2}+q$

We can now identify the values of $p$ and $q$ .
$p = 5$ $q = -44$
The equation $x^{2}-10x-19$ can be expressed as $(x-5)^{2}-44$ .