Write the quadratic equation whose roots are 6 and 1, and whose leading coefficient is 3.

Question

Loyau97 · Accepted Answer

Given: the roots are 1 and 6. That means,

x = 1 \Rightarrow (x - 1) = 0

and

x = 6 \Rightarrow (x - 6) = 0

Thus, a polynomial with leading coefficient a and roots 1 and 6 is

f (x) = a (x - 1) (x - 6)

.
Since the leading coefficient is 3,

a = 3

So, the required polynomial is:

f (x) = 3 (x - 1) (x - 6)

= 3 x^{2} - 21 x + 18

Vasquez · Accepted Answer

The quadratic equation with roots 6 and 1 and a leading coefficient of 3 can be written using the factored form. The factored form of a quadratic equation is given by

(x - r_{1}) (x - r_{2}) = 0

, where

r_{1}

and

r_{2}

are the roots. In this case, the equation becomes

(x - 6) (x - 1) = 0

.
To convert this equation into the standard form

a x^{2} + b x + c = 0

, we can expand the expression using the distributive property:

(x - 6) (x - 1) = 0

x^{2} - x - 6 x + 6 = 0

x^{2} - 7 x + 6 = 0

Since the leading coefficient is 3, we can multiply the entire equation by

\frac{1}{3}

to maintain the same roots and obtain the final equation:

\frac{1}{3} (x^{2} - 7 x + 6) = 0

\frac{1}{3} x^{2} - \frac{7}{3} x + 2 = 0

Therefore, the quadratic equation with roots 6 and 1 and a leading coefficient of 3 is

\frac{1}{3} x^{2} - \frac{7}{3} x + 2 = 0

.

Don Sumner · Accepted Answer

To write the quadratic equation with roots 6 and 1, we can use the fact that the roots of a quadratic equation in the form of

a x^{2} + b x + c = 0

are related to the coefficients of the equation through the following formulas:
The sum of the roots is given by the formula:

{Sum of roots} = \frac{- b}{a}

The product of the roots is given by the formula:

{Product of roots} = \frac{c}{a}

In our case, we are given that the roots are 6 and 1. So, the sum of the roots is 6 + 1 = 7, and the product of the roots is 6 * 1 = 6.
Using the formulas above, we can write the following equations:

\frac{- b}{a} = 7 {(1)}

\frac{c}{a} = 6 {(2)}

We are also given that the leading coefficient is 3, which means the coefficient of the quadratic term (

x^{2}

) is 3. Thus, we have:

a = 3 {(3)}

Now, let's solve equations (1) and (2) to find the values of

b

and

c

. Multiplying equation (1) by

a

and equation (2) by

a

, we get:

- b = 7 a {(4)}

c = 6 a {(5)}

Substituting the value of

a

from equation (3) into equations (4) and (5), we get:

- b = 7 \cdot 3 {(6)}

c = 6 \cdot 3 {(7)}

Simplifying equations (6) and (7), we have:

- b = 21 {(8)}

c = 18 {(9)}

Finally, we can write the quadratic equation using the values of

a

,

b

, and

c

we obtained:

3 x^{2} - 21 x + 18 = 0

This is the quadratic equation whose roots are 6 and 1, with a leading coefficient of 3.

nick1337 · Accepted Answer

Result:3x2−21x+18=0Solution:(x−r1)(x−r2)=0Substituting the given values r1=6 and r2=1, we have:(x−6)(x−1)=0Expanding the equation, we get:x2−7x+6=0Therefore, the quadratic equation with roots 6 and 1 and a leading coefficient of 3 is:3x2−21x+18=0

Write the quadratic equation whose roots are 6 and 1,

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