QuestionFind the coefficients $a$ , $b$ , and $c$ for the quadratic function with roots $x=2$ , $x=5$ , and passing through $(7,-40)$ .

Studdy Solution

STEP 1

Assumptions1. The roots of the quadratic function are $x=$ and $x=5$ . The function passes through the point $(7,-40)$ 3. The quadratic function is in the form $y=a x^{}+b x+c$

STEP 2

A quadratic function with roots $x=p$ and $x=q$ can be written in the form $y=a(x-p)(x-q)$ . We can use this to write the equation of our function.
$y=a(x-2)(x-5)$

STEP 3

We can expand this equation to get it in the form $y=ax^{2}+bx+c$ .
$y=a(x^{2}-7x+10)$

STEP 4

Expanding further, we get $y=ax^{2}-7ax+10a$ This gives us the coefficients of the quadratic function in terms of $a$ .

STEP 5

We know that the function passes through the point $(7,-40)$ . We can substitute these values into our equation to solve for $a$ .
$-40=a(7^{2})-7a(7)+10a$

STEP 6

implify the equation to solve for $a$ .
$-40=49a-49a+10a$

STEP 7

implify further to get $-40=10a$

STEP 8

olving for $a$ gives us $a=-40/10=-4$

STEP 9

Now that we have the value of $a$ , we can substitute it back into our expanded equation to find the values of $b$ and $c$ .
$y=-4x^{2}+28x-40$ From this equation, we can see that $b=28$ and $c=-40$ .
So, the values of $a, b$ , and $c$ are $-4,28$ , and $-40$ respectively.

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QuestionFind the coefficients $a$ , $b$ , and $c$ for the quadratic function with roots $x=2$ , $x=5$ , and passing through $(7,-40)$ .

Studdy Solution

STEP 1

Assumptions1. The roots of the quadratic function are $x=$ and $x=5$ . The function passes through the point $(7,-40)$ 3. The quadratic function is in the form $y=a x^{}+b x+c$

STEP 2

A quadratic function with roots $x=p$ and $x=q$ can be written in the form $y=a(x-p)(x-q)$ . We can use this to write the equation of our function.
$y=a(x-2)(x-5)$

STEP 3

We can expand this equation to get it in the form $y=ax^{2}+bx+c$ .
$y=a(x^{2}-7x+10)$

STEP 4

Expanding further, we get $y=ax^{2}-7ax+10a$ This gives us the coefficients of the quadratic function in terms of $a$ .

STEP 5

We know that the function passes through the point $(7,-40)$ . We can substitute these values into our equation to solve for $a$ .
$-40=a(7^{2})-7a(7)+10a$

STEP 6

implify the equation to solve for $a$ .
$-40=49a-49a+10a$

STEP 7

implify further to get $-40=10a$

STEP 8

olving for $a$ gives us $a=-40/10=-4$

STEP 9

Now that we have the value of $a$ , we can substitute it back into our expanded equation to find the values of $b$ and $c$ .
$y=-4x^{2}+28x-40$ From this equation, we can see that $b=28$ and $c=-40$ .
So, the values of $a, b$ , and $c$ are $-4,28$ , and $-40$ respectively.

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QuestionFind the coefficients aaa, bbb, and ccc for the quadratic function with roots x=2x=2x=2, x=5x=5x=5, and passing through (7,−40)(7,-40)(7,−40).

Studdy Solution

STEP 1

Assumptions1. The roots of the quadratic function are x=x=x= and x=5x=5x=5 . The function passes through the point (7,−40)(7,-40)(7,−40)3. The quadratic function is in the form y=ax+bx+cy=a x^{}+b x+cy=ax+bx+c

STEP 2

A quadratic function with roots x=px=px=p and x=qx=qx=q can be written in the form y=a(x−p)(x−q)y=a(x-p)(x-q)y=a(x−p)(x−q). We can use this to write the equation of our function.y=a(x−2)(x−5)y=a(x-2)(x-5)y=a(x−2)(x−5)

STEP 3

We can expand this equation to get it in the form y=ax2+bx+cy=ax^{2}+bx+cy=ax2+bx+c.y=a(x2−7x+10)y=a(x^{2}-7x+10)y=a(x2−7x+10)

STEP 4

Expanding further, we gety=ax2−7ax+10ay=ax^{2}-7ax+10ay=ax2−7ax+10aThis gives us the coefficients of the quadratic function in terms of aaa.

STEP 5

We know that the function passes through the point (7,−40)(7,-40)(7,−40). We can substitute these values into our equation to solve for aaa.−40=a(72)−7a(7)+10a-40=a(7^{2})-7a(7)+10a−40=a(72)−7a(7)+10a

STEP 6

implify the equation to solve for aaa.−40=49a−49a+10a-40=49a-49a+10a−40=49a−49a+10a

STEP 7

implify further to get−40=10a-40=10a−40=10a

STEP 8

olving for aaa gives usa=−40/10=−4a=-40/10=-4a=−40/10=−4

STEP 9

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QuestionFind the coefficients $a$ , $b$ , and $c$ for the quadratic function with roots $x=2$ , $x=5$ , and passing through $(7,-40)$ .

Assumptions1. The roots of the quadratic function are $x=$ and $x=5$ . The function passes through the point $(7,-40)$ 3. The quadratic function is in the form $y=a x^{}+b x+c$

A quadratic function with roots $x=p$ and $x=q$ can be written in the form $y=a(x-p)(x-q)$ . We can use this to write the equation of our function.
$y=a(x-2)(x-5)$

We can expand this equation to get it in the form $y=ax^{2}+bx+c$ .
$y=a(x^{2}-7x+10)$

Expanding further, we get $y=ax^{2}-7ax+10a$ This gives us the coefficients of the quadratic function in terms of $a$ .

We know that the function passes through the point $(7,-40)$ . We can substitute these values into our equation to solve for $a$ .
$-40=a(7^{2})-7a(7)+10a$

implify the equation to solve for $a$ .
$-40=49a-49a+10a$

implify further to get $-40=10a$

olving for $a$ gives us $a=-40/10=-4$