One of the roots of the equation x^2−6x+q=0 is 4.5. Find the other root and the value of the coefficient q.
h = - B/2A = 6/2 = 3 = axis of symmetry.
The roots are the same distance from axis of symmetry.
(1.5, 0), (3, 0), (4.5, 0). The roots are 1.5 and 4.5.
4.5^2 - 6*4.5 + q = 0
q =
x^2-6x+q = (x - 9/2)(x-a) = x^2 - (a + 9/2)x + 9/2 a
clearly, a + 9/2 = 6
now just finish it off
To find the other root of the equation, we can use the fact that the sum of roots of a quadratic equation is equal to the coefficient of the linear term divided by the coefficient of the quadratic term.
In this case, the sum of the roots is given by -b/a, where a is the coefficient of the quadratic term, and b is the coefficient of the linear term.
Since we have one root as 4.5, the sum of roots will be 4.5 plus the other root, which we'll call r. So, the sum of the roots is 4.5 + r.
Now we can equate this sum to the expression -b/a, which is -(-6)/1 in this case, since the coefficient of the linear term is -6 and the coefficient of the quadratic term is 1.
Therefore, we have:
4.5 + r = -(-6)/1
Let's simplify:
4.5 + r = 6
Now we can solve for r:
r = 6 - 4.5
r = 1.5
So, the other root of the equation is 1.5.
To find the value of the coefficient q, we can use the fact that the product of roots of a quadratic equation is equal to the constant term divided by the coefficient of the quadratic term.
In this case, the product of the roots is given by c/a, where a is the coefficient of the quadratic term, and c is the constant term.
The constant term in the equation x^2 − 6x + q = 0 is q. Therefore, the product of the roots is equal to q/1, since the coefficient of the quadratic term is 1.
Since we know one root is 4.5 and the other root is 1.5, the product of the roots is:
4.5 * 1.5 = 6.75
Therefore, the value of the coefficient q is 6.75.
To find the other root and the value of the coefficient q, we first need to understand the relationship between the roots of a quadratic equation and its coefficients.
For a quadratic equation of the form ax^2 + bx + c = 0, the sum of the roots is equal to -b/a, and the product of the roots is equal to c/a.
In this case, we have the equation x^2 - 6x + q = 0, and one of the roots is given as 4.5. Let's call the other root r.
Since the sum of the roots is equal to -b/a, we can write the equation as:
4.5 + r = -(-6)/1
Simplifying this equation, we get:
4.5 + r = 6
By subtracting 4.5 from both sides of the equation, we find:
r = 6 - 4.5 = 1.5
So the other root of the equation is 1.5.
Now, to find the value of the coefficient q, we can use the fact that the product of the roots is equal to c/a.
We know that one of the roots is 4.5, and the other root is 1.5. The product of these roots is (4.5)(1.5) = 6.75.
Since the coefficient of the x^2 term is 1, we can write the equation as:
1(6.75) = q
Therefore, q = 6.75.
In conclusion, the other root of the equation x^2 - 6x + q = 0 is 1.5, and the value of the coefficient q is 6.75.